It says that my federal income tax was filed for 2023 and paid. and I want to confirm that my Georgia state taxes were paid

Your only proof of payment is your bank statement or credit card showing the payment coming out.  Turbo Tax only passes your bank account number to the IRS or state for them to take it out of your account.  (Most states do not let you pay by Direct Debit so you have to mail in a check or go directly to the state's website to pay).  And some banks use different routing numbers for deposits and payments.  Double check the number you entered with your bank.

Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.

Only you would know if you paid your GA tax due.   Your own bank or credit card records are your only proof that you paid.  TurboTax never knows how, when or even if you paid your tax due.

TurboTax has no way to know if and or when taxes owed have been paid.

Your proof of payment will depend on how you decided to pay the taxes owed,  Payment by check, you would have a cancelled check.  Payment by direct debit from a bank account, you would have a bank statement showing the payment debited from your account.  Payment by credit card, you would have a credit card statement showing the payment debited from your account.