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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2016-09-12</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001195737</issuerCik>
        <issuerName>Nuveen AMT-Free Quality Municipal Income Fund</issuerName>
        <issuerTradingSymbol>NEA</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000072971</rptOwnerCik>
            <rptOwnerName>WELLS FARGO &amp; COMPANY/MN</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>420 MONTGOMERY STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>SAN FRANCISCO</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>94163</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001585457</rptOwnerCik>
            <rptOwnerName>Wells Fargo Municipal Capital Strategies, LLC</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>375 PARK AVENUE</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10152</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000740906</rptOwnerCik>
            <rptOwnerName>WELLS FARGO BANK N A</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>101 NORTH PHILLIPS STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>SIOUX FALLS</rptOwnerCity>
            <rptOwnerState>SD</rptOwnerState>
            <rptOwnerZipCode>57104</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Variable Rate MuniFund Term Preferred Shares</value>
            </securityTitle>
            <transactionDate>
                <value>2016-09-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>J</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
                <footnoteId id="F2"/>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>5350</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7730</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>By Subsidiary</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">In connection with the reorganization of the Nuveen Performance Plus Municipal Fund, Inc. into NEA, the disposition occurred in exchange for an equal number of shares of NEA, Series 2018, held by Wells Fargo Bank, National Association (&quot;WFBNA&quot;) of equivalent value in a cashless transaction.</footnote>
        <footnote id="F2">This statement is jointly filed by Wells Fargo &amp; Company, Wells Fargo Municipal Capital Strategies, LLC (&quot;Capital Strategies&quot;) and WFBNA.  Wells Fargo &amp; Company holds an indirect interest in the securities listed in Table I (the &quot;Securities&quot;) by virtue of its indirect ownership of its subsidiary Capital Strategies and WFBNA.  2,380 of the Securities are beneficially owned by Capital Strategies, and 5,350 of the Securities are beneficially owned by WFBNA.  Capital Strategies and WFBNA are each indirect wholly owned subsidiaries of Wells Fargo &amp; Company.</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>

    <remarks>Exhibits Index
Exhibit 99.1 - Joint Filing Agreement
Exhibit 99.2 - Joint Filer Information</remarks>

    <ownerSignature>
        <signatureName>WELLS FARGO &amp; COMPANY, / s / Michael Choquette</signatureName>
        <signatureDate>2016-09-14</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>WELLS FARGO MUNICIPAL CAPITAL STRATEGIES, LLC, / s / Adam Joseph</signatureName>
        <signatureDate>2016-09-14</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>WELLS FARGO BANK, NATIONAL ASSOCIATION, / s / Adam Joseph</signatureName>
        <signatureDate>2016-09-14</signatureDate>
    </ownerSignature>
</ownershipDocument>
