No, you will need TurboTax Deluxe to file your Form 1099-G
Nothing is really free from Intuit and Turbotax. Its very very greedy. If you make less then $66,000 you can get a free provider from the IRS website.
If you qualify for the IRS FreeFile program there is a separate TurboTax product for this. It can be found HERE but does require a separate login and password.
The commercial version of TurboTax is a paid product, with limited free options for the simplest of returns, but the free edition on the commercial website is not intended for taxpayers who would like to use the IRS sponsored FreeFile program.