Yes. According to the GA Department of Revenue, nonresidents, who work in Georgia or receive income from Georgia sources and are required to file a Federal income tax return, are required to file a Georgia income tax return. (If you have a net loss from this GA rental property, no GA state income tax return will be required.)
Some examples of Georgia source income are wages, Georgia lottery winnings, income from flow through entities (s-corporations, partnerships, LLC’s, trusts, and estates), rents, etc.
You will also need to file an IL resident state tax return (for all income from all sources including GA rental income). You will get a state income tax credit in IL for any GA state income taxes that you paid on your nonresident GA state income tax return.
You will want to work on your nonresident GA state income tax return first. You will then take a tax credit from your nonresident GA state income taxes on your resident IL state income tax return. (Please note that you will only get a tax credit for your GA state income taxes up to the amount of IL state income taxes that would have been paid if the income was earned in IL). The credit for taxes paid to another state section will be at the end of your residence state's interview process.
Just follow the TurboTax guide when working on your states (remembering to do your nonresident state return first) and TurboTax will do all the calculations and credits to your resident states return
Here is additional information about filing in multiple states (select "see more answer" to view the entire attachment)