If HFS (public aid) is involved, there is a good chance they will find out eventually. It may take some time, however,. Since you are expecting money, why not do the honorable thing and try to settle your arrearages?
Was a lawsuit filed regarding the accident? The State is not privy to your lawyer's demand letter and if the demand is made prior to a lawsuit being filed, it's unlikely that there will be any public record. Besides Illinois law allows a deduction for " medical expenditures necessary to preserve life or health" 750 ILCS 505(a)(3)(h). So if your settlement is for the purpose of repaying the doctors who treated you for injuries from the accident, it can't be touched.
The information provided here should not be construed to be formal legal advice. The provision of this general advice does not create a lawyer-client relationship. Persons with legal questions are encouraged to seek independent counsel for advice regarding their individual legal issues.