The 2021 state tax refund that you received in 2022 is taxable income on your federal tax return for 2022 only if you claimed an itemized deduction for state income tax on your 2021 federal tax return, and you actually got a tax benefit from deducting the amount that was later refunded. If you took the standard deduction on your 2021 federal tax return you can just ignore the state refund. Don't even bother entering it into TurboTax for 2022.
If you claimed itemized deductions on your 2021 federal tax return, enter the state refund in TurboTax and TurboTax will ask questions about your 2021 federal tax return to determine whether part or all of the refund is taxable. Because of the cap on the federal deduction for state and local taxes ("SALT") there is a good chance that the refund will turn out to not be taxable, but you have to let TurboTax do the calculations.
The state tax refund is not taxable income on your 2022 state tax return under any circumstances.
