Yes, you need to file a non-resident tax return for Arizona to claim the unemployment income.
You will then file a full-year resident state return for Illinois.
Illinois will tax you on all your income, but credit you for the tax you pay to Arizona, so please do the Arizona return first (so TurboTax can compute the Illinois credit).
You will need to contact the state of Arizona for a copy of the 1099-G, I assume it is available online.
**Say "Thanks" by clicking the thumb icon in a post
**Mark the post that answers your question by clicking on "Mark as Best Answer"