What state was the 1099-G from and do you have an amount in Box 2 for state or local refunds?
A general rule of thumb: If you didn’t deduct state and local income taxes last year, you don’t need to pay taxes on your state and local tax refund this year. For instance, if you didn’t itemize your deductions last year and instead you took the standard deduction, then your state tax refund from the previous year is tax-free this year. See Is my state tax refund taxable?