Can The State Push Child Support?

So, my boyfriend and I had our daughter. We don't live together and obviously not married. I do receive Food Stamps and Medical Card through Job and Family Services. We are very happy, plan on eventually moving in together and getting married, but not right now. I don't want to make him pay child support. My question is, will the state of Ohio push and make him pay it, where I do collect benefits?

TIA.