Why do I owe state taxes in Georgia when I did not make any income in the one month I lived there in 2023? Do I even need to file with the state if I did not have income?

If you did not receive any income while living in Georgia there is no reason to file a 2023 Georgia state tax return.

If you are a legal resident of Georgia for only a portion of the tax year and are required to file a federal income tax return, you are required to file a Georgia Form 500 Individual Income Tax Return. You must complete Schedule 3 to determine your Georgia Taxable income. A full explanation of how to compute the tax for part-year residents is provided in the IT-511 Individual Income Tax Booklet.

Reference: https://dor.georgia.gov/filing-requirements

If you had no GA income, your tax should be zero, but you are still required to file.  This is the rule, for part year residents, in most states