If you received a state income tax refund in 2019, you will enter it as income on your 2019 federal tax return, which you will file in 2020, no matter what year the refund was for. However, make sure that the state issued a 2019 Form 1099-G for the refund, not a 2018 1099-G. If they issued a 2018 1099-G, the simplest thing for you to do is report it as if you received it in 2018, rather than trying to get the state to correct the 1099-G.
Unfortunately, when the refund is not for the immediately preceding year, TurboTax will not calculate how much, if any, of the refund is taxable income. You do not have to report the 2017 state refund at all if you did not claim itemized deductions on your 2017 federal tax return, or if you claimed itemized deductions but deducted sales tax instead of state income tax. If you did claim an itemized deduction for state income tax, you would have to calculate the taxable amount of the refund yourself, following the instructions in IRS Publication 525, and enter the taxable amount in TurboTax for your 2019 tax return.
There are two tricks you can use to avoid having to calculate the taxable amount of the refund by hand. One is to simply report the full amount as taxable. There's a good chance that that would be the result anyway. You can use the second trick if you are using the CD/Download TurboTax software for your 2018 tax return. Keep the 2018 TurboTax software installed. Save a copy of your 2018 tax return file with a different name. (You can use Save As.) In the copy of your 2018 tax return, enter the state tax refund for 2018 and go through the calculation of the taxable amount. Then use that calculated taxable amount in your 2019 tax return. The taxable amount will be correct because the calculation only depends on information from your 2017 tax return. It doesn't matter what year you are reporting it in.
