Please advise your specific questions so we can better assist.

I’ve input all my taxes.  After paying 2019 EQT for both state and federal, I am set to get a fed return of nearly $500 and a state return of 1200.  Shouldn’t that come off my 2020 EQT?  I paid 869/qtr for fed and 512/qtr for state—will they just apply refunds to future 2020 EQT payments?   I put that I had paid EQT I. State and now my refund shows 0.  Will they print my new state and fed EQT amounts?

If you pay quarterly estimated taxes, apply a portion your refund to next year's first-quarter estimated tax payment, which is now due July 15. 

Click this link for more info on How to Apply Refund to Next Year's Taxes. 

TurboTax will prepare Estimated Tax Payment Vouchers for you, taking the refund amount you applied into account. 

Click this link for detailed info on Estimated Payments.