While the most common use of a Form 1099-G is to report unemployment compensation, there are also several other types of government payments that can be reported on a Form 1099-G, including state and local tax refunds.
Forms are not due until Jan. 31st so it is still possible that you will receive the form. However, what steps you should take depend on what reason you are expecting a Form 1099-G. If you received unemployment compensation, then you could likely find this information on your states labor department website.
If you didn't have a state or local tax refund last year, then you may not receive a Form 1099-G at all. If you did have a state refund, then you should also have this information in your tax or banking records from last year (it is just the refund amount that you received). Also, you are only required to actually input this information into TurboTax if you itemized your deductions last year. If you took the standard deduction, then your state or local tax refund will not be taxable income.
