You should consult with an attorney who is familiar with trust and estate administration to make certain that you need to pay this debt.
To answer your question, if the trust was a typical grantor retained revocable trust (a so called "living trust"), then "yes," they may be able to put a lien on the home. It depends on the loan agreement your father signed and state law regarding that type of debt, and the terms of the trust agreement.
It may be that the loan is a debt of your father's estate, and should be paid from estate assets. If there isn't enough money to pay the debts, then the loan company may have to settle for less than the full value of the debt. Additionally, living trusts often include language directing the trustee of the trust to pay the debts of the decedent's estate.
In any event, you should consult a trust and estates attorney licensed to practice law in the jurisdiction of your father's domicile (most likely, where he lived). In addition to answering this question for you, a trust and estates attorney will help you with the overall administration of your father's trust and estate and help you make certain that the administration is done correctly -- particularly when it comes to paying debts like the one your asked about.
Best of luck.