You only need to include in your federal return your state or local tax refund(s), reported in Box 2 of Form 1099-G, if you had taken a deduction on your federal return for paying those state or local taxes in a prior year. Box 3 of Form 1099-G will indicate the relevant tax year. The itemized deduction for state and local taxes paid is claimed on Form 1040 Schedule A Line 5a, State and Local Income Taxes.
To enter your Form 1099-G information into TurboTax, please see the folllowing TurboTax FAQ, "Where do I enter a 1099-G for a state or local tax refund?":