Legal Question in Family Law in Illinois

If no legal matter has been filed by her or on her behalf, then there is no restriction on her leaving the state. If she or HFS has filed suit against Father # 1, then she needs court permission to leave. Be aware that Father # 1 may file suit after she leaves and request that the child be returned to Illinois. If he does not see the child and has relationship with the child, he has a difficult case. If he sees the child regularly, then he has a decent chance of obtaining a visitation order of some regularity.

Techically, all custodial parents (such as your girlfriend) is to ask for permission to leave the state with a minor child. In the real world people do it all the time and nothing happens unless the non custodial parent complains. I would recommend asking the court for permission to leave the state as it 1. is the law, and 2. avoids you having to fight a court battle in Illinois after you have moved to TX.