No, you do not need to amend the federal tax return.
If you never worked or lived in Georgia in 2021 then any state income tax paid to Georgia are eligible to be refunded to you on the 2021 amended state tax return.
See this TurboTax support FAQ for how to amend a state tax return - https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-return/need-amend-state-return/L3hGF...
