<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0206</schemaVersion>

    <documentType>3</documentType>

    <periodOfReport>2019-03-14</periodOfReport>

    <noSecuritiesOwned>0</noSecuritiesOwned>

    <issuer>
        <issuerCik>0001181024</issuerCik>
        <issuerName>BLACKROCK NEW YORK MUNICIPAL INCOME QUALITY TRUST</issuerName>
        <issuerTradingSymbol>BSE</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000070858</rptOwnerCik>
            <rptOwnerName>BANK OF AMERICA CORP /DE/</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>BANK OF AMERICA CORPORATE CENTER</rptOwnerStreet1>
            <rptOwnerStreet2>100 N TRYON ST</rptOwnerStreet2>
            <rptOwnerCity>CHARLOTTE</rptOwnerCity>
            <rptOwnerState>NC</rptOwnerState>
            <rptOwnerZipCode>28255</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001565450</rptOwnerCik>
            <rptOwnerName>Banc of America Preferred Funding Corp</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>214 NORTH TRYON STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>CHARLOTTE</rptOwnerCity>
            <rptOwnerState>NC</rptOwnerState>
            <rptOwnerZipCode>28255</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Variable Rate Demand Preferred Shares</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>405</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                    <footnoteId id="F2"/>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>By Subsidiary</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">The 405 variable rate demand preferred shares (&quot;Shares&quot;) reported as acquired in Table I represent Shares of BlackRock New York Municipal Income Quality Trust (the &quot;Issuer&quot;) beneficially owned by Banc of America Preferred Funding Corporation (&quot;PFC&quot;).  The Shares were acquired for a purchase price of $100,099.561654 per share (representing $100,000 per share liquidation preference and accrued dividends of $99.561654 per share).  PFC is a wholly owned subsidiary of Bank of America Corporation (&quot;Bank of America&quot;).</footnote>
        <footnote id="F2">This statement is jointly filed by Bank of America and PFC.  Bank of America holds an indirect interest in the securities listed in Table I (the &quot;Securities&quot;) by virtue of its indirect ownership of its subsidiary PFC.</footnote>
        <footnote id="F3">Each reporting person declares that neither the filing of this statement nor anything herein shall be construed as an admission that such person is, for the purposes of Section 13(d) of the US Securities Exchange Act of 1934 or any other purpose, (i) acting (or has agreed or is agreeing to act together with any other person) as a partnership, limited partnership, syndicate, or other group for the purpose of acquiring, holding or disposing of securities of the Issuer or otherwise with respect to the Issuer or any securities of the Issuer or (ii) a member of any group with respect to the Issuer or any securities of the Issuer.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Ronnie Ojera (Bank of America Corporation)</signatureName>
        <signatureDate>2019-03-25</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Michael Jentis (Banc of America Preferred Funding Corporation)</signatureName>
        <signatureDate>2019-03-25</signatureDate>
    </ownerSignature>
</ownershipDocument>
