how do I fight the state of nj

If the house is owned by you - I don't know how anyone could put a lien on the house unless you signed for whatever debt mom is incurring.  The Caretaker Child exemption becomes important in getting a person Medicaid eligible.  If there was a gap between eligibility transfer of the house, and Medicaid, your mom might be responsible for those costs.  But still, I don't know how a lien was imposed with out a judgment or some other court proceeding.