<PAGE>

                                                                    EXHIBIT 10

                         BAY VIEW CAPITAL CORPORATION

                                     AND

            SUNTRUST BANK, CENTRAL FLORIDA, NATIONAL ASSOCIATION


                        FIRST SUPPLEMENTAL INDENTURE

                        Dated as of October 30, 1998



                         SUPPLEMENTING AND AMENDING 

                                     THE

                                  INDENTURE

                         Dated as of August 28, 1997



                        SUBORDINATED DEBT SECURITIES




<PAGE>
 
        FIRST SUPPLEMENTAL INDENTURE, dated as of October 30, 1998 (the "First
Supplemental Indenture"), between Bay View Capital Corporation, a Delaware 
corporation, (the "Issuer" which term as used herein includes any successor 
thereto), and SunTrust Bank, Central Florida, National Association, a national
banking association, as trustee (the "Trustee").


                           RECITALS OF THE ISSUER

        WHEREAS, the Issuer and the Trustee have duly authorized, executed and
delivered a certain indenture dated as of August 28, 1997 (the "Indenture") 
and an Officers Certificate pursuant to Section 201 and 301 of the Indenture 
(the "Officer's Certificate") to secure and to provide for the authentication 
and delivery of the Issuer's Subordinated Debt Securities (the "Notes"), 
including an initial series of Notes known as the Issuer's "9 1/8% 
Subordinated Notes due 2007"; and

        WHEREAS, the Issuer and the Trustee desire to amend the Indenture as 
hereinafter provided;

        NOW, THEREFORE, in consideration of the premises and of the covenants 
herein contained and of other good and valuable consideration, the receipt and
sufficiency of which are hereby acknowledged, the Issuer and the Trustee 
hereby agree as follows:

        Section 1.  For all purposes of this First Supplemental Indenture, 
        ---------
unless the context otherwise requires, the capitalized terms used herein shall
have the meanings assigned to them in the Indenture.

        Section 2.  Clause (b)(11)(a)(A) of the Officer's Certificate is 
        ---------
hereby amended in its entirety as follows:

        declare, pay or set apart any funds for the payment of any dividend
        on, or make any distribution to holders of, any Capital Stock of the
        Company or any Capital Stock of any Subsidiary of the Company (other
        than (i) payments to the Company or a Subsidiary of the Company, (ii)
        dividends or distributions to the extent payable in Qualified Capital
        Stock of the Company and (iii) dividends on preferred securities
        issued by a Subsidiary whose purpose is primarily to invest all the
        proceeds from the issuance of such preferred securities in debt
        securities of the Company and any distributions of such debt securities 
        to the holders of the preferred securities).

        Section 3.  The definition of Regulatory Capital Requirement contained
        ---------
in Clause (b)(12) of the Officer's Certificate is hereby amended in its 
entirety as follows:

        "Regulatory Capital Requirements" means (i) with respect to the
        Company, the minimum amount of capital required for the Company
        (assuming for such purpose that the Company is subject to regulation
        as a bank holding company under federal law, whether or not the
        Company is in fact so subject) to be deemed to be

                                      2
<PAGE>
 
        "adequately capitalized" (or the then equivalent category) under
        federal banking laws and regulations as then in effect and applicable
        to the Company (assuming it is subject to regulation as a bank holding
        company as aforesaid) at such time; (ii) with respect to any Principal
        Bank Subsidiary of the Company at any time, the minimum amount of
        capital required for such Principal Bank Subsidiary to be deemed to be
        "well capitalized" (or the then equivalent category) or, solely for
        purposes of clause (c) of the first paragraph under (b)(11)(d) hereof
        (Limitation on Consolidation, Merger and Sale of Assets), "adequately
        capitalized" (or the then equivalent category) under federal banking
        laws and regulations as then in effect and applicable to such Principal
        Bank Subsidiary; (iii) with respect to any Successor Person, the
        minimum amount of capital required for such Successor Person (assuming
        for such purpose that such Successor Person is subject to regulation
        as a bank holding company under federal law, whether or not it is in
        fact so subject) to be deemed to be "adequately capitalized" (or the
        then equivalent category) under federal banking laws and regulations
        as then in effect and applicable to such Successor Person (assuming it
        is subject to regulation as a bank holding company as aforesaid) at
        such time; and (iv) with respect to any Principal Bank Subsidiary of
        such Successor Person at any time, the minimum amount of capital
        required for such Principal Bank Subsidiary to be deemed to be "well
        capitalized" (or the then equivalent category) or, solely for purposes
        of clause (c) of the first paragraph under (b)(11)(d) hereof
        (Limitation on Consolidation, Merger and Sale of Assets), "adequately
        capitalized" (or the then equivalent category) under federal banking
        laws and regulations as then in effect and applicable to such
        Principal Bank Subsidiary, at such time.

        Section 4.  Except as specifically amended or supplemented by this 
        ---------
First Supplemental Indenture, all provisions of the Indenture shall remain 
applicable for all purposes, and the Indenture is hereby ratified, confirmed 
and approved. The Indenture and this First Supplemental Indenture shall 
together be construed as one and the same instrument.

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<PAGE>
 
        IN WITNESS WHEREOF, the parties have caused this First Supplemental 
Indenture to be duly executed as of the day and year first above written.

                                  BAYVIEW CAPITAL CORPORATION,         
                                  a Delaware Corporation                     
                                                                             
                                  By: /s/ EDWARD H. SONDKER                  
                                      ----------------------------------------
                                  Name: Edward H. Sondker                    
                                        --------------------------------------
                                  Title: President and Chief Executive Officer
                                         -------------------------------------

[SEAL]

Attest:

/s/ ROBERT J. FLAX
-------------------------------------
Name: Robert J. Flax
Title: Executive Vice President,
       General Counsel and Secretary


                                  SUNTRUST BANK, CENTRAL FLORIDA,
                                  NATIONAL ASSOCIATION, as Trustee

                                  By: /s/ LISA DERRYBERRY
                                      ------------------------
                                  Name: Lisa Derryberry
                                        ----------------------
                                  Title: Vice President
                                         ---------------------

[SEAL]

Attest:

/s/ PATRICIA SMITH
------------------------------
Name: Patricia Smith
Title: Corporate Trust Officer



                                      4
<PAGE>
 
    OFFICER'S CERTIFICATE
    ---------------------

    Each of the undersigned hereby certify that they are duly elected and 
incumbent officers of Bay View Capital Corporation ("BVCC"), a Delaware 
corporation. In order for Silver Freedman & Taff, LLP to render certain opinions
in connection with the proposed First Supplemental Indenture, to be dated the 
date of this certificate (the "Supplemental Indenture"), between BVCC and 
SunTrust Bank, Central Florida, National Association, as trustee (the "Trustee")
the undersigned hereby certifies the accuracy of the facts contained herein, and
to the best of our knowledge make the representations provided herein. The First
Supplemental Indenture would amend the Indenture (the "Indenture") and an 
Officers' Certificate Pursuant to Sections 201 and 301 of the Indenture (the 
"Officers' Certificate"), each dated August 28, 1997, between BVCC and the 
Trustee, which govern the terms and conditions of BVCC's 9 1/8% Subordinated 
Notes Due August 15, 2007 in the aggregate principal amount of $100,000,000 (the
"Notes"). Capitalized terms used but not defined herein shall have the meanings 
assigned to such terms in the Indenture. Each of the undersigned further 
certifies to the Trustee and Silver Freedman & Taff, LLP for reliance in issuing
the legal opinion as follows:

1.  BVCC will only issue additional debt securities in reliance on amended 
Clause (b)(11)(a)(A) of the Officer's Certificate to a Subsidiary whose sole 
purpose is to invest all the net proceeds from the Subsidiary's issuance of 
preferred securities in debt securities of the BVCC and engaging only in 
activities necessary and advisable or incidental thereto.

2.  BVCC will record on its books and records and report on its financial 
statements the preferred securities issued by the Subsidiary as a liability of 
BVCC for financial accounting purposes in accordance with generally accepted 
accounting principles and as debt of BVCC for all tax purposes.

3.  Neither the sale of the debt securities to the Subsidiary by BVCC will occur
at a time when the insolvency of BVCC is contemplated nor a trustee in 
bankruptcy has been appointed for BVCC. BVCC will not be rendered insolvent by 
the sale of the debt securities to the Subsidiary. After giving effect to the 
sale of the debt securities, BVCC will have sufficient capital to support its 
business and BVCC at the time of issuance of the debt securities has available 
funds and anticipated earnings that are reasonably anticipated to be sufficient 
to meet its ongoing expenses. BVCC has no intent to issue debt securities beyond
its ability to pay its debts and expenses on a timely basis as they come due.

4.  The amendments do not adversely affect the interests of the Holders of the 
Notes in any material respect.


                         BAY VIEW CAPITAL CORPORATION
<TABLE> 
<CAPTION> 
        Officer                 Name                            Signature
        -------                 ----                            ---------
<S>                     <C>                             <C>
President and Chief     Edward H. Sondker               /s/ EDWARD H. SONDKER
Executive Officer                                       ------------------------

Executive Vice          David A. Heaberlin              /s/ DAVID A. HEABERLIN
President, Chief                                        ------------------------
Financial Officer
and Treasurer
</TABLE> 



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