No, you do not include/add information reported on Form 1099-G to your W-2 entries. On the entry screen for your 1099-G, there is a check box for "Federal or state income tax was withheld on this payment." Check that box. The next screen will be for entering tax withholding reported on your 1099-G.
Note: Your state's version of the 1099-G might have different box numbers than what is shown in TurboTax. If your state's numbers don't match ours, make sure you are entering the correct information.
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