<PAGE>   1

                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549



                                    Form 10-Q



                Quarterly Report Pursuant To Section 13 or 15(d)
                     of the Securities Exchange Act of 1934



                  For the Quarterly Period Ended March 31, 1998


                           Commission File No. 1-8032



                          SAN JUAN BASIN ROYALTY TRUST


Texas                                                    I.R.S. No. 75-6279898


                     Bank One, Texas, N.A., Trust Department
                                 P. O. Box 2604
                             Fort Worth, Texas 76113

                          Telephone Number 817/884-4630



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 and (2) has been subject to such filing requirements for
the past 90 days. Yes  X    No
                      ---      ---

Number of units of beneficial interest outstanding at May 15, 1998:   46,608,796



                                  Page 1 of 12


<PAGE>   2

                          SAN JUAN BASIN ROYALTY TRUST

                         PART I - FINANCIAL INFORMATION


Item 1.  Financial Statements

The condensed financial statements included herein have been prepared by Bank
One, Texas, N.A. as Trustee for the San Juan Basin Royalty Trust, without audit,
pursuant to the rules and regulations of the Securities and Exchange Commission.
Certain information and footnote disclosures normally included in annual
financial statements have been condensed or omitted pursuant to such rules and
regulations, although the Trustee believes that the disclosures are adequate to
make the information presented not misleading. It is suggested that these
condensed financial statements be read in conjunction with the financial
statements and the notes thereto included in the Trust's latest annual report on
Form 10-K. In the opinion of the Trustee, all adjustments, consisting only of
normal recurring adjustments, necessary to present fairly the assets,
liabilities and trust corpus of the San Juan Basin Royalty Trust at March 31,
1998, and the distributable income and changes in trust corpus for the
three-month periods ended March 31, 1998 and 1997 have been included. The
distributable income for such interim periods is not necessarily indicative of
the distributable income for the full year.

Deloitte & Touche LLP, independent certified public accountants, has made a
review of the condensed financial statements as of March 31, 1998 and for the
three-month periods ended March 31, 1998 and 1997 included herein.



                                      - 2 -


<PAGE>   3

INDEPENDENT ACCOUNTANTS' REPORT


Bank One, Texas, N.A. as Trustee for the
   San Juan Basin Royalty Trust:

We have reviewed the accompanying condensed statement of assets, liabilities and
trust corpus of the San Juan Basin Royalty Trust as of March 31, 1998 and the
related condensed statements of distributable income and changes in trust corpus
for the three-month periods ended March 31, 1998 and 1997. These financial
statements are the responsibility of the Trustee.

We conducted our review in accordance with standards established by the American
Institute of Certified Public Accountants. A review of interim financial
information consists principally of applying analytical procedures to financial
data and making inquiries of persons responsible for financial and accounting
matters. It is substantially less in scope than an audit conducted in accordance
with generally accepted auditing standards, the objective of which is the
expression of an opinion regarding the financial statements taken as a whole.
Accordingly, we do not express such an opinion.

The accompanying condensed financial statements are prepared on a modified cash
basis as described in Note 1, which is a comprehensive basis of accounting other
than generally accepted accounting principles.

Based on our review, we are not aware of any material modifications that should
be made to such condensed financial statements for them to be in conformity with
the basis of accounting described in Note 1.

We have previously audited, in accordance with generally accepted auditing
standards, the statement of assets, liabilities and trust corpus of the San Juan
Basin Royalty Trust as of December 31, 1997, and the related statements of
distributable income and changes in trust corpus for the year then ended (not
presented herein); and in our report dated March 25, 1998, we expressed an
unqualified opinion on those financial statements. In our opinion, the
information set forth in the accompanying condensed statement of assets,
liabilities and trust corpus as of December 31, 1997 is fairly stated in all
material respects in relation to the statement of assets, liabilities and trust
corpus from which it has been derived.





DELOITTE & TOUCHE LLP

May 8, 1998




                                     - 3 -
<PAGE>   4

SAN JUAN  BASIN ROYALTY TRUST

CONDENSED STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS
-------------------------------------------------------------------------------

<TABLE>
<CAPTION>
                                                                          MARCH 31,         DECEMBER 31,
ASSETS                                                                       1998              1997
                                                                         (UNAUDITED)

<S>                                                                      <C>                <C>
Cash and short-term investments                                          $ 3,664,248        $ 5,111,832
Net overriding royalty interests in producing oil and 
  gas properties (net of accumulated amortization of 
  $78,662,298 and $77,156,080 at March 31, 1998
  and December 31, 1997, respectively)                                    54,613,230         56,119,448
                                                                         -----------        -----------

                                                                         $58,277,478        $61,231,280
                                                                         ===========        ===========
LIABILITIES AND TRUST CORPUS

Distribution payable to Unit holders                                     $ 3,664,248        $ 5,111,832
Commitments and contingencies
Trust corpus - 46,608,796 Units of beneficial
   interest authorized and outstanding                                    54,613,230         56,119,448
                                                                         -----------        -----------

                                                                         $58,277,478        $61,231,280
                                                                         ===========        ===========
</TABLE>

CONDENSED STATEMENTS OF DISTRIBUTABLE INCOME (UNAUDITED)
--------------------------------------------------------------------------------


<TABLE>
<CAPTION>
                                                                                THREE MONTHS ENDED
                                                                                     MARCH 31,
                                                                         --------------------------------
                                                                             1998               1997

<S>                                                                      <C>                 <C>
Royalty income                                                           $ 11,663,131        $ 18,471,262
Interest income                                                                28,048              27,657
                                                                         ------------        ------------
                                                                           11,691,179          18,498,919

General and administrative expenditures                                       249,207             231,502
                                                                         ------------        ------------

Distributable income                                                     $ 11,441,972        $ 18,267,417
                                                                         ============        ============

Distributable income per Unit (46,608,796 Units)                         $    .245489        $    .391930
                                                                         ============        ============
</TABLE>


The accompanying notes to condensed financial statements are an integral part of
these statements.





                                      - 4 -
<PAGE>   5

CONDENSED STATEMENTS OF CHANGES IN TRUST CORPUS (UNAUDITED)
--------------------------------------------------------------------------------


<TABLE>
<CAPTION>
                                                                                THREE MONTHS ENDED
                                                                                     MARCH 31,
                                                                         ---------------------------------
                                                                              1998                1997

<S>                                                                      <C>                  <C>
Trust corpus, beginning of period                                        $ 56,119,448         $ 62,808,148
Amortization of net overriding royalty interest                            (1,506,218)          (2,170,937)
Distributable income                                                       11,441,972           18,267,417
Distributions declared                                                    (11,441,972)         (18,267,417)
                                                                         ------------         ------------

Trust corpus, end of period                                              $ 54,613,230         $ 60,637,211
                                                                         ============         ============
</TABLE>


The accompanying notes to condensed financial statements are an integral part of
this statement.



                                      - 5 -

<PAGE>   6


SAN JUAN BASIN ROYALTY TRUST


NOTES TO CONDENSED FINANCIAL STATEMENTS (UNAUDITED)
--------------------------------------------------------------------------------

1.    BASIS OF ACCOUNTING

      The San Juan Basin Royalty Trust ("Trust") was established as of November
      1, 1980. The financial statements of the Trust are prepared on the
      following basis:

      o  Royalty income recorded for a month is the amount computed and paid by
         the working interest owner, Burlington Resources Oil & Gas ("BROG"), to
         the Trustee for the Trust. Royalty income consists of the amounts
         received by the owner of the interest burdened by the net overriding
         royalty interest ("Royalty") from the sale of production less accrued
         production costs, development and drilling costs, applicable taxes,
         operating charges, and other costs and deductions, multiplied by 75%.

      o  Trust expenses recorded are based on liabilities paid and cash reserves
         established from royalty income for liabilities and contingencies.

      o  Distributions to Unit holders are recorded when declared by the
         Trustee.

      o  The conveyance which transferred the overriding royalty interest to the
         Trust provides that any excess of production costs over gross proceeds
         must be recovered from future net profits.

      The financial statements of the Trust differ from financial statements
      prepared in accordance with generally accepted accounting principles
      ("GAAP") because revenues are not accrued in the month of production;
      certain cash reserves may be established for contingencies which would not
      be accrued in financial statements prepared in accordance with GAAP; and
      amortization of the Royalty calculated on a unit-of-production basis is
      charged directly to trust corpus.

2.    FEDERAL INCOME TAXES

      For Federal income tax purposes, the Trust constitutes a fixed investment
      trust which is taxed as a grantor trust. A grantor trust is not subject to
      tax at the trust level. The Unit holders are considered to own the Trust's
      income and principal as though no trust were in existence. The income of
      the Trust is deemed to have been received or accrued by each Unit holder
      at the time such income is received or accrued by the Trust rather than
      when distributed by the Trust.

      The Royalty constitutes an "economic interest" in oil and gas properties
      for Federal income tax purposes. Unit holders must report their share of
      the revenues of the Trust as ordinary income from oil and gas royalties
      and are entitled to claim depletion with respect to such income. The
      Royalty is treated as a single property for depletion purposes.

      The Trust has on file technical advice memoranda confirming the tax
      treatment described above.

      The Trust began receiving royalty income from coal seam wells beginning in
      1989. Under Section 29 of the Internal Revenue Code, production from coal
      seam gas wells drilled prior to January 1, 1993, qualifies for the federal
      income tax credit for producing non-conventional fuels. This tax credit
      was approximately $1.05 per MMBtu for the year 1997 and is adjusted for
      inflation annually. The credit currently applies to production through the
      year 2002. Production from wells drilled after



                                      - 6 -
<PAGE>   7

      December 31, 1979, but prior to January 1, 1993, to a formation beneath a
      qualifying coal seam formation which are later completed into such
      formation also qualifies for the tax credit. Each Unit holder must
      determine his pro rata share of such production based upon the number of
      Units owned during each month of the year and apply the tax credit against
      his own income tax liability, but such credit may not reduce his regular
      tax liability (after the foreign tax credit and certain other
      nonrefundable credits) below his tentative minimum tax. Section 29 also
      provides that any amount of Section 29 credit disallowed for the tax year
      solely because of this limitation will increase his credit for prior year
      minimum tax liability, which may be carried forward indefinitely as a
      credit against the taxpayer's regular tax liability, subject, however, to
      the limitations described in the preceding sentence. There is no provision
      for the carryback or carryforward of the Section 29 credit in any other
      circumstances.

      The classification of the Trust's income for purposes of the passive loss
      rules may be important to a Unit holder. As a result of the Tax Reform Act
      of 1986, royalty income will generally be treated as portfolio income and
      will not reduce passive losses.

                                     ******


                                      - 7 -


<PAGE>   8

ITEM 2. TRUSTEE'S DISCUSSION AND ANALYSIS

FORWARD LOOKING INFORMATION

Certain information included in this report contains, and other materials filed
or to be filed by the Trust with the Securities and Exchange Commission (as well
as information included in oral statements or other written statements made or
to be made by the Trust) may contain or include, forward looking statements
within the meaning of Section 21E of the Securities Exchange Act of 1934, as
amended, and Section 27A of the Securities Act of 1933, as amended. Such forward
looking statements may be or may concern, among other things, capital
expenditures, drilling activity, development activities, production efforts and
volumes, hydrocarbon prices and the results thereof, and regulatory matters.
Such forward looking statements generally are accompanied by words such as
"estimate," "expect," "predict," "anticipate," "goal," "should," "assume,"
"believe," or other words that convey the uncertainty of future events or
outcomes.

YEAR 2000 ISSUE

As the year 2000 approaches, there are uncertainties concerning whether computer
systems will properly recognize date-sensitive information when the year changes
to 2000. Systems that do not properly recognize such information could generate
erroneous data or fail. The Trust is in communication with third parties with
which it deals, but it is not yet possible to fully assess the effect of a third
party's inability to adequately address year 2000 issues.

THREE MONTHS ENDED MARCH 31, 1998 AND 1997

During the first quarter of 1998, the San Juan Basin Royalty Trust received
royalty income of $11,663,131. Distributable income consists of royalty income
plus interest income less administrative expenses. Interest income for the
quarter was $28,048 and administrative expenses were $249,207. Thus,
distributable income totaled $11,441,972. Based on 46,608,796 Units outstanding,
the per Unit distributions during the first quarter of 1998 were as follows:

<TABLE>
<S>                                                <C>      
     January                                       $ .095019
     February                                        .071853
     March                                           .078617
                                                   ---------

     Quarter Total                                 $ .245489
                                                   =========
</TABLE>

The amount distributed to Unit holders in the first quarter of 1998 was less
than the $.391930 distributed in the first quarter of 1997. The decrease was
primarily attributable to a decrease in the average gas price from $3.05 per Mcf
for the first quarter of 1997 to $2.08 per Mcf for the first quarter of 1998.
The tax credit relating to production from coal seam wells totaled approximately
$.05 per Unit for the first quarter of 1998 compared to $.06 per Unit for the
first quarter of 1997. For further information concerning this tax credit, Unit
holders should refer to the Trust's Annual Report for 1997.

Interest income for the first quarter of 1998 was more than the $27,657 in the
first quarter of 1997 due to higher interest rates. Administrative expenses of
$249,207 were higher in the first three months of 1998 as compared to $231,502
in the first quarter of 1997, primarily as a result of differences in timing of
the receipt and payment of these expenses.

Capital expenditures incurred by BROG, attributable to the properties from which
the Royalty was carved ("Underlying Properties"), for the first quarter of 1998
amounted to $2,263,043. Capital expenditures were $2,320,206 for the first
quarter of 1997.



                                     - 8 -
<PAGE>   9
 In the first quarter of 1998, four gross (1.73 net) conventional wells were
completed on the Underlying Properties. Two gross (1.75 net) conventional wells
were recompleted and nine gross (.37 net) coal seam wells were recavitated
through March 31, 1998. There were five gross (1.60 net) conventional wells, six
gross (2.02 net) conventional recompletions, one gross (.04 net) coal seam well,
five gross (.22 net) coal seam recompletions and nine gross (.36 net) coal seam
recavitations in progress on March 31, 1998. In the first quarter of 1997, 13
gross (.51 net) coal seam wells were completed on the properties from which the
Royalty was carved. One gross (.84) coal seam well was recompleted and 4 gross
(.55 net) coal seam wells were recavitated through March 31, 1997. There was one
gross (.83 net) conventional well recompleted through March 31, 1997. There were
13 gross (2.15 net) conventional wells, 1 gross (.87 net) conventional well
recompletion, 1 gross (.04 net) coal seam well and 4 gross (.16 net)
recompletions to Fruitland Coal from other formations in progress on March 31,
1997. Unit holders are referred to "Description of the Properties" in the
Trust's Annual Report for 1997 for further information concerning BROG's coal
seam well drilling program in the San Juan Basin. This program includes
properties in which the Trust owns an interest.

Royalty income for the quarter ended March 31, 1998 is associated with actual
gas and oil production during November 1997 through January 1998 from the
properties from which the Royalty was carved. Gas and oil sales for the quarters
ended March 31, 1998 and 1997 were as follows:

<TABLE>
<CAPTION>
                                                                      1998             1997

<S>                                                                <C>              <C> 
PROPERTIES FROM WHICH THE ROYALTY WAS CARVED:
Gas:
   Total sales (Mcf)                                               10,961,460       10,613,520
   Mcf per day                                                        119,146          115,364
   Average price (per Mcf)                                       $       2.08            $3.05

Oil:
   Total sales (Bbls)                                                  20,472           20,962
   Bbls per day                                                           223              228
   Average price (per Bbl)                                             $16.10      $     22.64

SALES ATTRIBUTABLE TO THE ROYALTY:
Gas sales (Mcf)                                                     6,168,933        6,542,089
Oil sales (Bbls)                                                       11,560           12,663
</TABLE>

Coal seam gas production decreased from 5,020,207 Mcf in the first quarter of
1997 to 4,959,361 Mcf in the first quarter of 1998. The price paid for gas
production decreased during the first quarter of 1998 primarily due to lower
index prices in the San Juan Basin. Gas sales volumes attributable to the
Underlying Properties increased overall in the first quarter of 1998 primarily
due to the increased volumes from conventional wells.

The price received per barrel of oil in the first quarter of 1998 was lower than
that received in the first quarter of 1997 due to decreases in the posted
prices. Lease operating expenses and property taxes totaled $2,945,740 during
the first quarter of 1998 compared to $2,887,013 for the first quarter of 1997.

Sales volumes attributable to the Royalty are determined by dividing the net
profits received by the Trust and attributable to oil and gas, respectively, by
the prices received for sales volumes from the Underlying Properties, taking
into consideration production taxes attributable to the Underlying Properties.

Since the gas and oil sales attributable to the Royalty are based on an
allocation formula that is dependent on such factors as price and cost
(including capital expenditures), the production volumes from the Underlying
Properties



                                     - 9 -
<PAGE>   10
do not provide a meaningful comparison to volumes attributable to the Royalty.

Effective January 1, 1998, all volumes of Trust gas became subject to the terms
of a Natural Gas Sales and Purchase Contract between BROG and El Paso Energy
Marketing Company ("El Paso"). That contract is for a term of two years through
and including December 31, 1999, and provides for the sale of Trust gas at
prices which will fluctuate in accordance with published indices for gas sold in
the San Juan Basin of New Mexico. BROG entered into the contract with El Paso
after soliciting and receiving competitive bids in late 1997 from six major gas
marketing firms to market and/or purchase the Trust gas. Unit holders are
referred to Note 6 of the Notes to Financial Statements in the Trust's 1997
annual Report for further information concerning the marketing of gas produced
from the Underlying Properties.

CALCULATION OF ROYALTY INCOME

Royalty income received by the Trust for the three months ended March 31, 1998
and 1997 was computed as shown in the following table:

<TABLE>
<CAPTION>
                                                                              1998                1997

<S>                                                                      <C>                 <C>         
Gross proceeds of sales from the Underlying Properties:
   Gas proceeds                                                          $ 22,793,991        $ 32,404,871
   Oil proceeds                                                               329,680             474,537
                                                                         ------------        ------------

Total                                                                      23,123,671          32,879,408
                                                                         ------------        ------------

Less production costs:
   Severance tax - Gas                                                      2,324,745           2,994,864
   Severance tax - Oil                                                         39,302              48,975
   Lease operating expense and property tax                                 2,945,740           2,887,013
   Capital expenditures                                                     2,263,043           2,320,206
                                                                         ------------        ------------

Total                                                                       7,572,830           8,251,058
                                                                         ------------        ------------

Net profits                                                                15,550,841          24,628,350
Net overriding royalty interest                                                    75 %                75 %
                                                                         ------------        ------------


Royalty income                                                           $ 11,663,131        $ 18,471,262
                                                                         ============        ============
</TABLE>



ITEM 3. QUALITATIVE AND QUANTITATIVE DISCLOSURES ABOUT MARKET RISK

Not applicable.



                                     - 10 -


<PAGE>   11

                           PART II - OTHER INFORMATION


Item 1.        Legal Proceedings

                     The Trust is not a party to any litigation. However, the
               Trust is aware that BROG is involved in litigation from time to
               time that could affect the royalty income received by the Trust.

                     A lawsuit has been commenced against BROG by certain
               royalty and overriding royalty owners on behalf of those persons
               similarly situated. The suit involves properties that are
               burdened by the Trust. This case is one of six virtually
               identical class actions filed against New Mexico gas producers.
               All such cases have been consolidated in the First Judicial
               District of Santa Fe County, New Mexico where the case is styled
               San Juan 1990-A,L.P.,et al. v. El Paso Production Company and
               Meridian Oil Inc. The plaintiffs allege that they and members of
               the proposed class have been underpaid for royalties and
               overriding royalties. The plaintiffs have sought to certify the
               actions as class actions and seek monetary damages. The court has
               denied class certification. Because of the pending nature of the
               litigation, exposure to the Trust from this suit cannot be
               quantified. However, if the plaintiffs who have interests in
               properties that are burdened by the Trust are successful, royalty
               income received by the Trust could decrease.

                      In addition, the Trust is aware of an administrative
               claim brought by the Mineral Management Service of the United
               States Department of the Interior (the "MMS") against BROG
               regarding a gas contract settlement agreement dated March 1,
               1990, between BROG and certain other parties thereto. The claim
               alleges that additional royalties are due on production from
               federal and Indian leases in the State of New Mexico on
               properties that are burdened by the Trust. If the MMS claim is
               successful, royalty income received by the Trust could decrease.
  
Items 2-5      Not Applicable

Item 6.        Exhibits and Reports on Form 8-K

               (a)   Exhibits

                     (4)(a)    San Juan Basin Royalty Trust Indenture dated
                               November 3, 1980, between Southland Royalty
                               Company and The Fort Worth National Bank (now
                               Bank One, Texas, N.A.), as Trustee, heretofore
                               filed as Exhibit (4)(a) to the Trust's Annual
                               Report on Form 10-K to the Securities and
                               Exchange Commission for the fiscal year ended
                               December 31, 1980 is incorporated herein by
                               reference.

                     (4)(b)    Net Overriding Royalty Conveyance from Southland
                               Royalty Company to The Fort Worth National bank
                               (now Bank One, Texas, N.A.), as Trustee, dated
                               November 3, 1980 (without Schedules), heretofore
                               filed as Exhibit (4)(b) to the Trust's Annual
                               Report on Form 10-K to the Securities and
                               Exchange Commission for the fiscal year ended
                               December 31, 1980 is incorporated herein by
                               reference.

                     (27)      Financial Data Schedule

               (b)   No reports on Form 8-K have been filed during the quarter
                     for which this report is filed.




                                     - 11 -


<PAGE>   12
                                   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.

                                  BANK ONE, TEXAS, N.A. AS TRUSTEE FOR THE SAN 
                                  JUAN BASIN ROYALTY TRUST


                                            By  /s/ LEE ANN ANDERSON
                                               -------------------------------
                                                       Lee Ann Anderson
                                                        Vice President

Dated as of May 15, 1998


               (The Trust has no directors or executive officers.)




                                     - 12 -

<PAGE>   13

                                INDEX TO EXHIBITS


<TABLE>
<CAPTION>
                                                                                                        SEQUENTIALLY
               EXHIBIT                                                                                     NUMBERED
               NUMBER                                EXHIBIT                                                PAGE

<S>                       <C>                                                                           <C>
               (4)(a)     San Juan Basin Royalty Trust Indenture dated November 3, 1980, between
                          Southland Royalty Company and The Fort Worth National Bank (now Bank One,
                          Texas, N.A.), as Trustee, heretofore filed as Exhibit (4)(a) to the
                          Trust's Annual Report on Form 10 K to the Securities and Exchange
                          Commission for the fiscal year ended December 31, 1980 is incorporated
                          herein by reference.*


               (4)(b)     Net Overriding Royalty Conveyance from Southland Royalty Company to The
                          Fort Worth National Bank (now Bank One, Texas, N.A.), as Trustee, dated
                          November 3, 1980 (without Schedules), heretofore filed as Exhibit (4)(b)
                          to the Trust's Annual Report on Form 10-K to the Securities and Exchange
                          Commission for the fiscal year ended December 31, 1980 is incorporated
                          herein by reference.*

               (27)       Financial Data Schedule **
</TABLE>


*    A copy of this Exhibit is available to any Unit holder, at the actual cost
     of reproduction, upon written request to the Trustee, Bank One, Texas,
     N.A., P.O. Box 2604, Fort Worth, Texas 76113.
**   Filed herewith.