Can a quit claim property in a trust with a mortgage be distributed to a designated heir? A recording by the grantor said not to pay it off

There are really not enough facts to completely answer this question.  There should be some language in the trust that will indicate when, or under what conditions the property can be distributed.  if those conditions are met, the property could be distributed, but then the beneficiary might have to take the property with the lien of the mortgage, and then be responsible for paying it.