According to the Georgia Department of Revenue, a Georgia state tax return is required of all Georgia part-year residents, if they are otherwise filing a federal income tax return. Unlike some other states that have more lax filing standards for part-year residents, Georgia is clear in that everyone must file if also filing a federal return). You can confirm this for yourself here, at the following government webpage:
https://dor.georgia.gov/filing-requirements
However, just because you are legally requried to file a tax return in Georgia, doesn't mean that you will actually owe any taxes there; and it doesn't mean you need to use TurboTax to create this return.
If you feel up to the challenge, you can download a blank copy of the Georgia state Form 500, which includes instructions for part-year residents, independently of the TurboTax software. You can then print whatever documents you need, complete them by hand, and then mail them. Here are courtesy links to the relevant Georgia Department of Revenue webpages:
https://dor.georgia.gov/sites/dor.georgia.gov/files/related_files/document/TSD/Form/2016%20IT-511%20...
https://dor.georgia.gov/documents/forms?field_form_type_tid=2826
(You would want the IT-511 packet.)
It may take some time, but once you have your federal tax return information data from TurboTax, creating a Georgia return may not be as difficult as it may seem at the moment. And, this option is free (no Georgia state charge from TurboTax).
Regardless of what you do for Georgia, you should still file a federal tax return, as well as a part-year resident state tax return for wherever you lived prior to moving into Georgia (assuming that this state has a personal income tax system; some states, like Florida, do not).
Thank you for asking this important question.
