Maybe. The fact that your refund was offset to repay unemployment benefits that you owed has no effect on its taxability; you constructively received the refund. The determination of whether the refund is taxable income is a function of whether you itemized deductions and obtained a tax benefit in the year represented by the refund.
Please follow these steps:
You should report last year's state or local tax refund. TurboTax will figure out if it's taxable or not.
If you took the state and local tax deduction last year and itemized, the refund will be taxable this year if the deduction benefitted (reduced) last year's tax bill. This is true even if you opted to apply the refund to this year's state or local return.
But if you took the sales tax deduction or the standard deduction last year, it won't be taxable on this year's return. It also won't be taxable if last year's return was subject to the AMT (Alternative Minimum Tax) and the refund amount was less than the amount disallowed under AMT.
After you enter last year's state or local refund in TurboTax, we'll ask followup questions to determine if it's taxable.