The way I am reading your post seems to be that you never completed a G-4 for tax year 2023. If that is the case then you need to pay GA estimated taxes on or before January 16, 2024. Or you can wait and file the 2023 state return and pay the taxes with the return and any underpayment penalties assessed on the return.
Go to this Georgia website to initiate a state estimated tax payment - https://georgia.gov/pay-estimated-tax
If you are going to continue to work for this Georgia employer then you should complete a G-4 starting now for tax year 2024.
