For the State of Georgia, you need to file a Form G-1003 Withholding Income Statement Transmittal and Form DOL-4A Annual Tax and Wage Report for Domestic Employment to account for the wages paid to and tax withheld from your domestic employee. It is likely your payroll service filed those returns, but you should check with them to make sure.
You do not report any tax with your Georgia state personal income tax return as far as your domestic employee wages are concerned like you do on your federal return. Consequently, you don't need to account for the state tax withholding from your domestic employee on your personal tax return for Georgia.
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