My relative passed away while residing in a nursing home and on Title 19, but had property. They had money left in a checking account and a trust account.  What happens to that money?

Your relative's estate must go through Probate Court to settle the estate.  A family member may petition the court by completing an application, PC-200.  On that application you will indicate what assets they had at the time of their passing including any real estate owned.  That application goes to Probate for review then to the Department of Administrative Services (DAS) Collections Department for review as well. 

DAS Collections would investigate to see if they owed money to the State of Connecticut’s State Medicaid.  If it is discovered that there were monies owed to the State of Connecticut, the estate will be liquidated to pay that debt.