No, you cannot add the 2019 1099-G to your 2020 return. However, you may not have to amend your 2019 tax return if it was a state or local income tax refund received in 2019 and you did not have any benefit from deducting it in prior years. If you do not itemize your deductions, you never received a tax benefit from the tax paid and therefore, do not have to amend your tax return for the refund.
If it was for unemployment and never entered, the IRS may have automatically adjusted your refund or sent you a bill for tax due. If that did not occur and it was income that should be reported, it would be best to amend your return.
If you have to amend your 2019 return, please go to How to amend (change or correct) a return you already filed. For 2019 amendments in TurboTax, scroll down to Your tax return & documents (you may need to select Show), select 2019 and then Amend (change) return. From there, you can make your changes. When finished, TurboTax will give you instructions for printing and mailing your amended return as amendments can't be e-filed.