<PAGE>   1
                                                        EXHIBIT INDEX ON PAGE 13


                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q


[XX]     QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15 (d) OF THE SECURITIES
         EXCHANGE ACT OF 1934

For the quarterly period ended:   MARCH 31, 1998

                                       or

[_]      TRANSITION REPORT PURSUANT TO SECTION 13 OR 15 (d) OF THE SECURITIES
         EXCHANGE ACT OF 1934

For the transition period from ___________________ to ___________________


Commission File Number:   1-6064

                               ALEXANDER'S, INC.
             (Exact name of registrant as specified in its charter)


              DELAWARE                                     51-0100517
(State or other jurisdiction of incorporation          (I.R.S. Employer
            or organization)                         Identification Number)


PARK 80 WEST, PLAZA II, SADDLE BROOK, NEW JERSEY             07663
    (Address of principal executive offices)               (Zip Code)


                                 (201)587-8541
              (Registrant's telephone number, including area code)


      (Former name, former address and former fiscal year, if changed since
                                  last report)

                 Indicate by check mark whether the registrant (1) has filed all
reports required to be filed by Section 13 or 15(d) of the Securities Exchange
Act of 1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.
                                                  [X] Yes    [ ] No

                APPLICABLE ONLY TO ISSUERS INVOLVED IN BANKRUPTCY
                   PROCEEDINGS DURING THE PRECEDING FIVE YEARS

                 Indicate by check mark whether the registrant has filed all
reports required to be filed by Sections 12, 13 or 15(d) of the Securities
Exchange Act of 1934 subsequent to the distribution of securities under a plan
confirmed by a court.
                                                  [ ] Yes    [ ] No

        As of May 1, 1998 there were 5,000,850 common shares outstanding.



                                  Page 1 of 14
<PAGE>   2
                                ALEXANDER'S, INC.

                                      INDEX



<TABLE>
<CAPTION>
                                                                                                     Page Number
                                                                                                     -----------

<S>                <C>                                                                               <C>    
PART I.            FINANCIAL INFORMATION:

        Item 1.    Financial Statements:

                   Consolidated Balance Sheets as of March 31, 1998
                   and December 31, 1997...........................................................      3

                   Consolidated Statements of Operations for the Three
                   Months Ended March 31, 1998 and March 31, 1997..................................      4

                   Consolidated Statements of Cash Flows for the Three
                   Months Ended March 31, 1998 and March 31, 1997..................................      5

                   Notes to Consolidated Financial Statements......................................      6

        Item 2.    Management's Discussion and Analysis of Financial
                   Condition and Results of Operations.............................................      8


PART II.           OTHER INFORMATION:


        Item 6.    Exhibits and Reports on Form 8-K................................................     11


Signatures         ................................................................................     12

Exhibit Index      ................................................................................     13

Exhibit 27         ................................................................................     14
</TABLE>


                                  Page 2 of 14
<PAGE>   3
PART I.  FINANCIAL INFORMATION

                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                           CONSOLIDATED BALANCE SHEETS
                   (AMOUNTS IN THOUSANDS EXCEPT SHARE AMOUNTS)

<TABLE>
<CAPTION>
                                                     MARCH 31,       DECEMBER 31,
                                                        1998             1997
                                                     ---------        ---------
<S>                                                  <C>              <C>      
ASSETS:

Real estate, at cost:
     Land                                            $  45,571        $  45,571
     Buildings, leaseholds and improvements            123,612          123,612
     Capitalized expenses and predevelopment
         costs                                          50,767           47,163
                                                     ---------        ---------
              Total                                    219,950          216,346
     Less accumulated depreciation and
         amortization                                  (35,862)         (35,224)
                                                     ---------        ---------
                                                       184,088          181,122

     Investment in unconsolidated joint
         venture                                        10,708           10,611
                                                     ---------        ---------
     Real estate, net                                  194,796          191,733

Cash and cash equivalents                                2,700            2,691
Restricted cash                                          8,209            1,872
Receivable arising from condemnation
     proceedings                                            --           14,700
Accounts receivable, net of allowance for
     doubtful accounts of $147 in each period              475            1,064
Receivable arising from the straight-lining
     of rents, net                                       8,703            7,805
Deferred lease and other expenses                       12,672           12,443
Deferred debt expense                                      632              783
Other assets                                             3,912            1,983
                                                     ---------        ---------

TOTAL ASSETS                                         $ 232,099        $ 235,074
                                                     =========        =========

LIABILITIES AND STOCKHOLDERS'
     EQUITY:

Debt                                                 $ 204,359        $ 208,087
Amounts due to Vornado Realty Trust and
     its affiliate                                       6,686            6,888
Accounts payable and accrued liabilities                 4,265            4,174
Other liabilities from discontinued operations           2,238            2,296
Minority interest                                          600              600
                                                     ---------        ---------

     TOTAL LIABILITIES                                 218,148          222,045
                                                     ---------        ---------

Commitments and contingencies

Stockholders' Equity:

Common stock; $1.00 par value per share;
     authorized 10,000,000 shares;
     issued 5,173,450                                    5,174            5,174
Additional capital                                      24,843           24,843
Deficit                                                (15,106)         (16,028)
                                                     ---------        ---------
                                                        14,911           13,989
Less treasury shares, 172,600 shares
     at cost                                              (960)            (960)
                                                     ---------        ---------
     Total stockholders' equity                         13,951           13,029
                                                     ---------        ---------


TOTAL LIABILITIES AND STOCKHOLDERS'
     EQUITY                                          $ 232,099        $ 235,074
                                                     =========        =========
</TABLE>


                 See notes to consolidated financial statements.
<PAGE>   4
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                      CONSOLIDATED STATEMENTS OF OPERATIONS


(amounts in thousands except share amounts)
<TABLE>
<CAPTION>
                                                       FOR THE THREE MONTHS ENDED
                                                       --------------------------
                                                         MARCH 31,      MARCH 31,
                                                           1998           1997
                                                         -------        -------
<S>                                                      <C>            <C>    
Revenues:
    Property rentals                                     $ 5,631        $ 4,220
    Expense reimbursements                                   998            491
    Equity in income of unconsolidated
       joint venture                                       1,378          1,287
                                                         -------        -------
Total revenues                                             8,007          5,998
                                                         -------        -------

Expenses:
    Operating (including management fee to Vornado
       of $210 in 1998 and 1997)                           2,020          1,612
    General and administrative (including
       management fee to Vornado of $540
       in 1998 and 1997)                                     866            981
    Depreciation and amortization                            798            571
                                                         -------        -------
Total expenses                                             3,684          3,164
                                                         -------        -------

Operating income                                           4,323          2,834

Interest and debt expense
    (including interest on loan from Vornado)             (3,665)        (3,294)
Interest and other income, net                               264            252
                                                         -------        -------

Net income (loss)                                        $   922        $  (208)
                                                         =======        =======

Net income (loss) per share (basic and diluted)          $   .18        $  (.04)
                                                         =======        =======
</TABLE>


                See notes to consolidated financial statements.


                                  Page 4 of 14
<PAGE>   5
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                      CONSOLIDATED STATEMENTS OF CASH FLOWS

(amounts in thousands)
<TABLE>
<CAPTION>
                                                                    FOR THE THREE MONTHS ENDED
                                                                     MARCH 31,       MARCH 31,
                                                                       1998            1997
                                                                     --------        --------
<S>                                                                  <C>             <C>      
CASH FLOWS FROM OPERATING ACTIVITIES:
  Net income (loss)                                                  $    922        $   (208)
  Adjustments to reconcile net income (loss) to net
    cash provided by (used in) operating activities:
      Depreciation and amortization
        (including debt issuance costs)                                 1,169           1,057
      Straight-lining of rental income                                   (898)           (512)
  Change in assets and liabilities:
      Accounts receivable                                                 589             (64)
      Note receivable                                                  14,700              --
      (Investment) distributions in excess of equity in income
        of unconsolidated joint venture                                   (98)          1,063
      Amounts due to Vornado Realty Trust and its affiliate              (527)           (542)
      Accounts payable and accrued liabilities                             92             527
      Other liabilities from discontinued operations                      (58)           (249)
      Other                                                            (1,994)         (1,552)
                                                                     --------        --------


Net cash provided by (used in) operating activities                    13,897            (480)
                                                                     --------        --------

CASH FLOWS FROM INVESTING ACTIVITIES:
  Additions to real estate                                             (3,603)         (4,398)
  Cash restricted for construction and development                     (6,326)              2
  Cash restricted for operating liabilities                               (11)            579
                                                                     --------        --------
Net cash (used in) investing activities                                (9,940)         (3,817)
                                                                     --------        --------

CASH FLOWS FROM FINANCING ACTIVITIES:
  Issuance of debt                                                         --          16,667
  Debt repayments                                                      (3,729)           (329)
  Deferred debt expense                                                  (219)           (167)
                                                                     --------        --------
Net cash (used in) provided by financing activities                    (3,948)         16,171
                                                                     --------        --------

Net increase in cash and cash equivalents                                   9          11,874
Cash and cash equivalents at beginning of period                        2,691           5,480
                                                                     --------        --------

Cash and cash equivalents at end of period                           $  2,700        $ 17,354
                                                                     ========        ========

SUPPLEMENTAL DISCLOSURE OF CASH FLOW INFORMATION:
  Cash payments for interest (including capitalized
    interest of $1,765 and $2,254)                                   $  5,060        $  5,063
                                                                     ========        ========
</TABLE>




                See notes to consolidated financial statements.

                                  Page 5 of 14
<PAGE>   6
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS




1.  CONSOLIDATED FINANCIAL STATEMENTS

      The consolidated balance sheet as of March 31, 1998, the consolidated
    statements of operations for the three months ended March 31, 1998, and the
    consolidated statements of cash flows for the three months ended March 31,
    1998 are unaudited. In the opinion of management, all adjustments (which
    include only normal recurring adjustments) necessary to present fairly the
    financial position, results of operations and changes in cash flows have
    been made.

      Certain information and footnote disclosures normally included in
    financial statements prepared in accordance with generally accepted
    accounting principles have been condensed or omitted. These condensed
    consolidated financial statements should be read in conjunction with the
    financial statements and notes thereto included in the Company's 1997 Annual
    Report to Shareholders. The results of operations for the three months ended
    March 31, 1998 are not necessarily indicative of the operating results for
    the full year.


2.  RELATIONSHIP WITH VORNADO REALTY TRUST ("Vornado")

      Vornado owns 29.3% of the Company's Common Stock. The Company is managed
    by and its properties are redeveloped and leased by Vornado, pursuant to
    agreements with a one-year term expiring in March of each year which are
    automatically renewable. Under these agreements, the Company incurred fees
    of $1,563,000 in each of the three month periods ended March 31, 1998 and
    1997. In addition, Vornado is due $6,078,000 at March 31, 1998 under the
    leasing agreement, subject to the payment of rents by tenants.

      Vornado lent the Company $45,000,000, the subordinated tranche of a
    $75,000,000 loan, in 1995. The loan, which had a three-year term expiring on
    March 15, 1998, has been extended for an additional year and the interest
    rate has been reset from 15.60% per annum to 13.87% per annum. The Company
    incurred interest on its loan from Vornado of $ 1,718,000 and $1,830,000 in
    the three months ended March 31, 1998 and 1997, of which $907,000 and
    $1,216,000 was capitalized.


3.  COMMITMENTS AND CONTINGENCIES

      Lexington Avenue

      The Company is evaluating redevelopment plans for this site, which may
    involve razing the existing buildings (in which case, the carrying cost of
    approximately $15,000,000 would be written-off) and developing a large
    multi-use building requiring capital in excess of $300,000,000 to be
    expended. No development decisions have been finalized.


                                  Page 6 of 14
<PAGE>   7
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS


      Environmental Matters

      In June 1997, the Kings Plaza Shopping Center (the "Center"), in which the
    Company owns a 50% interest, commissioned an Environmental Study and
    Contamination Assessment Site Investigation (the Phase II "Study") to
    evaluate and delineate environmental conditions disclosed in a Phase I
    study. The results of the Study indicate the presence of petroleum and other
    hydrocarbons in the soil and groundwater. The Study recommends a remedial
    approach, but agreement has not yet been reached with the New York State
    Department of Environmental Conservation ("NYDEC") on the finalization of
    the approach. The Center accrued $1,500,000 at December 31, 1997, for its
    estimated obligation with respect to the clean up of the site, which
    includes costs of (i) remedial investigation, (ii) feasibility study, (iii)
    remedial design, (iv) remedial action and (v) professional fees. If the
    NYDEC insists on a more extensive remediation approach, the Center could
    incur additional obligations.

      Such contamination may have resulted from activities of third parties;
    however, the sources of the contamination have not been fully identified.
    Although the Center intends to pursue all available remedies against any
    potentially responsible third parties, there can be no assurance that such
    parties will be identified, or if identified, whether these potentially
    responsible third parties will be solvent. In addition, the costs associated
    with pursuing any potentially responsible parties may be cost prohibitive.
    The Center has not recorded an asset as of March 31, 1998 for potential
    recoveries of environmental remediation costs from other parties.

      Compliance with applicable provisions of federal, state and local laws
    regulating the discharge of materials into the environment or otherwise
    relating to the protection of the environment have not had, and, although
    there can be no assurance, are not expected to have, a material effect on
    the Company's operations, earnings, competitive position or capital
    expenditures.

4.  EARNINGS PER SHARE

    The following table set for the computation of basic and diluted earnings
  per share:

                                                    FOR THE THREE MONTHS ENDED
<TABLE>
<CAPTION>
                                                   ----------------------------
                                                    MARCH 31,         MARCH 31,
                                                       1998              1997
                                                   ----------       -----------
<S>                                                <C>              <C>         
Numerator:
  Net income (loss)                                $  922,000       $  (208,000)
                                                   ==========       ===========

Denominator:
  Denominator for basic earnings per share --
     weighted average shares                        5,000,850         5,000,850
  Effect of dilutive securities:
     Employee stock options                            67,335                --
                                                   ----------       -----------

  Denominator for diluted earnings per share --
     adjusted weighted average shares and
     assumed conversions                            5,068,185         5,000,850
                                                   ==========       ===========

Basic and diluted earnings (loss) per share        $      .18       $      (.04)
                                                   ==========       ===========
</TABLE>


                                  Page 7 of 14
<PAGE>   8
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS


RESULTS OF OPERATIONS

         The Company's revenues, which consist of property rentals, tenant
expense reimbursements and equity in income of unconsolidated joint venture were
$8,007,000 in the quarter ended March 31, 1998, compared to $5,998,000 in the
quarter ended March 31, 1997, an increase of $2,009,000 or 33.5%.

         Property rentals were $5,631,000 in the quarter ended March 31, 1998,
compared to $4,220,000 in the quarter ended March 31, 1997, an increase of
$1,411,000 or 33.4%. This increase resulted primarily from rents from leases
commencing subsequent to March 31, 1997 at the Rego Park I and Kings Plaza Store
properties of $2,117,000, partially offset by the loss of $965,000 of rent
resulting from Caldor's rejection of its Fordham Road lease in June 1997.

         Tenant expense reimbursements were $998,000 in the quarter ended March
31, 1998, compared to $491,000 in the quarter ended March 31, 1997, an increase
of $507,000. This increase reflects a corresponding increase in operating
expenses passed through to tenants as a result of leases commencing subsequent
to March 31, 1997 at the Rego Park I property and the commencement of operations
at the Kings Plaza Store property.

         Equity in income of unconsolidated joint venture (the Kings Plaza Mall)
was $1,378,000 in the quarter ended March 31, 1998, compared to $1,287,000 in
the quarter ended March 31, 1997, an increase of $91,000 or 7.1%. This increase
resulted primarily from an increase in rent from mall tenants.

         Operating expenses were $2,020,000 in the quarter ended March 31, 1998,
compared to $1,612,000 in the quarter ended March 31, 1997, an increase of
$408,000. This increase resulted primarily from real estate taxes which
previously had been capitalized, being charged to income due to the 
commencement of operations at the Kings Plaza Store property.

         General and administrative expenses were $866,000 in the quarter ended
March 31, 1998, compared to $981,000 in the quarter ended March 31, 1997, a
decrease of $115,000. This decrease resulted primarily from lower professional
fees.

         Depreciation and amortization expense was $798,000 in the quarter ended
March 31, 1998, compared to $571,000 in the quarter ended March 31, 1997, an
increase of $227,000. This increase resulted primarily from the commencement of
operations at the Kings Plaza Store property in October 1997.

         Interest and debt expense was $3,665,000 in the quarter ended March 31,
1998, compared to $3,294,000 in the quarter ended March 31, 1997, an increase of
$371,000. This increase resulted primarily from interest allocated to the Kings
Plaza Store property which previously had been capitalized, being charged to 
income in 1998.

LIQUIDITY AND CAPITAL RESOURCES

         In the aggregate, Alexander's current operating properties (six of its
nine properties) do not generate sufficient cash flow to pay all of its
expenses. The Company's three non-operating properties (Lexington Avenue,
Paramus, and Rego Park II) are in various stages of redevelopment. As rents
commence from a portion of the redevelopment properties, the Company expects
that cash flow will become positive.


                                  Page 8 of 14
<PAGE>   9
                                ALEXANDER'S, INC.

                                AND SUBSIDIARIES

                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS

         The Company estimates that its capital expenditure requirements for
redevelopment projects will include: (i) the redevelopment of the Paramus
property, at a cost of approximately $90,000,000 to $100,000,000 and (ii)
improvements to the Kings Plaza Shopping Center at a cost of approximately
$20,000,000. Further, the Company is evaluating redevelopment plans for the
Lexington Avenue site, which may involve razing the existing buildings (in which
case, the carrying cost of approximately $15,000,000 would be written-off) and
developing a large multi-use building requiring capital in excess of
$300,000,000 to be expended. While the Company anticipates that financing will
be available after tenants have been obtained for these redevelopment projects,
there can be no assurance that such financing will be obtained or if obtained,
that such financings will be on terms that are acceptable to the Company. In
addition, it is uncertain as to when these projects will commence.

         Property rentals from Caldor, which filed for relief under Chapter 11
of the United States Bankruptcy Code in September 1995, represented
approximately 12% and 22% of the Company's consolidated revenues for the three
months ended March 31, 1998 and for the year ended December 31, 1997. Caldor
rejected its Fordham Road lease effective June 6, 1997 and accordingly, no
longer pays rent. Alexander's has filed a claim for damages based on such
rejection. The annual base rental revenue under this lease was $3,537,000. The
loss of property rental payments, if any, under the Caldor lease for the
Flushing property, could have a material adverse affect on the Company's
financial condition and results of operations.

         The non-affiliated limited partners of the Seven Thirty One Limited
partnership (the "Partnership"), a limited partnership which owns the Company's
Lexington Avenue property, have the right to put their remaining 7.64% interest
to the Partnership until October 3, 1998, in exchange for a five year secured
note in the principal amount of $15,000,000, bearing annual interest at Prime
plus 1%.

         The Company estimates that the fair market values of its assets are
substantially in excess of their historical cost and that it has additional
borrowing capacity. Alexander's continues to evaluate its needs for capital
which may be raised through (a) property specific or corporate borrowing, (b)
the sale of securities and (c) asset sales.

    In addition, in March 1998 the Company received the proceeds from the
condemnation of a portion of its Paramus property of $14,700,000 and repaid
$3,591,000 of the loan collateralized by the property.

    Although there can be no assurance, the Company believes that these cash
sources will be adequate to fund cash requirements until its operations generate
adequate cash flow.

    CASH FLOWS

    Three Months Ended March 31, 1998
    Cash provided by operating activities of $13,897,000 was comprised of (i)
net income of $922,000, (ii) proceeds from the condemnation of a portion of the
Paramus property of $14,700,000 and (iii) non-cash items of $271,000, offset by
(iv) the net change in operating assets and liabilities of $1,996,000. The
adjustments for non-cash items are comprised of depreciation and amortization of
$1,169,000, offset by the effect of straight-lining of rental income of
$898,000.

    Net cash used in investing activities of $9,940,000 was primarily comprised
of (i) cash from the condemnation of a portion of the Paramus property
restricted for construction and development of $6,326,000 and (ii) capital
expenditures of $3,603,000.

    Net cash used in financing activities of $3,948,000 was comprised primarily
of repayments of debt.


                                  Page 9 of 14
<PAGE>   10
                                ALEXANDER'S, INC.
                                AND SUBSIDIARIES

                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS



    Three Months Ended March 31, 1997
    Cash used in operating activities of $480,000 was comprised of (i) a net
loss of $208,000, (ii) the payment of liabilities of discontinued operations of
$249,000 and (iii) the net change in operating assets and liabilities of
$568,000, offset by adjustments for non-cash items of $545,000. The adjustments
for non-cash items are comprised of depreciation and amortization of $1,057,000,
offset by the effect of straight-lining of rental income of $512,000.

    Net cash used in investing activities of $3,817,000 was comprised of capital
expenditures of $4,398,000, offset by the release of cash restricted for both
operating liabilities and construction financing of $581,000.

    Net cash provided by financing activities of $16,171,000 was comprised of
proceeds from the issuance of debt (net of deferred debt expense) of $16,500,000
on the Rego Park I property, offset by repayments of debt of $329,000.



                                  Page 10 of 14
<PAGE>   11
                                ALEXANDER'S, INC.



        PART II.    OTHER INFORMATION


              ITEM 6.    EXHIBITS AND REPORTS ON FORM 8-K.

                    (a)  Exhibits:  The following exhibits are filed with this
                                           Quarterly Report on Form 10-Q.


                         27  Financial Data Schedule

                    (b)  Reports on Form 8-K

                         None




                                  Page 11 of 14
<PAGE>   12
                                ALEXANDER'S, INC.


                                   SIGNATURES





              Pursuant to the requirements of the Securities Exchange Act of
1934, the registrant has duly caused this report to be signed on its behalf by
the undersigned thereunto duly authorized.




                                                  ALEXANDER'S, INC.
                                           ------------------------------------
                                                    (Registrant)
                                        
                                        
                                        
Date:  May 7, 1998                                 /s/ Joseph Macnow
                                           ------------------------------------
                                                     JOSEPH MACNOW
                                             Vice President - Chief Financial
                                           Officer and Chief Accounting Officer









                                  Page 12 of 14
<PAGE>   13
                                ALEXANDER'S, INC.

                                  EXHIBIT INDEX


                                                                PAGE NUMBER IN
                                                                   SEQUENTIAL
         EXHIBIT NO.                                                NUMBERING
         -----------                                            --------------




            27         Financial Data Schedule                           14







                                  Page 13 of 14