Get your taxes done using TurboTax

Correct. (but I think you mean you needed to pay estimated 2024 local tax)

Usually the state refund is for the previous tax year, so the process (to claim or not to claim the 1099-G) is a bit complicated, but since you're doing the same federal tax year as the refund, just lower the amount of local tax you claim on your 2024 federal return. 

Under

Deductions & Credits

Estimates and other Taxes Paid

Estimates 

Local estimated taxes for 2024

You can enter the estimated payments made in this section, and if you hadn't gotten a refund, you would report all the payments you made in this section. Since you DID get a refund, only report the difference in this section.

If you paid 500, and got 100 refund, simply enter 400.