It depends. It sounds like after receiving a state tax refund for 2023 in 2024, you received a 2024 1099-G. If that is correct, and you itemized your deductions last year (and didn't choose the sales tax deduction in lieu of income tax deduction), then yes, you will be taxed on the refund.
However, if you took the standard deduction in 2023, or deducted sales tax, then no, you do not need to report your 2023 state tax refund, and you have no responsibility to pay any tax on the refund.
Here is an article you may find helpful: Is My State Tax Refund Taxable?