On my Illinois return - Education expense credit-eligible dependents asks if your child was in between k-12th. my child graduated in 2022 and began college still qualify?

Your child was a high school student in 2022.  You can enter expenses related to attending high school on the IL state form.  You cannot enter college expenses.

Yes.

If you are referring to the Illinois Tax Credit for education expenses, your dependent would qualify since they were a student in grade K-12 in 2022 and under the age of 21. 

Since you mention that the student started college in the Fall of 2022, that dependent would also qualify as a college student for any Federal education credits you may be eligible for. 

According to the State of Illinois:

“The Illinois Education Expense Credit allows parents or legal guardians of a fulltime K-12 student under the age of 21, to take a 25% tax credit on qualified education expenses over $250 this year. The total credit may not exceed $750, regardless of the number of qualifying students.”

IL