When filling out a Quit Claim Deed form to transfer property into my living trust with me as the trustee, do I put me as the grantee in the deed form?

A quit claim deed does not transfer title.  It merely announces that someone quits making a claim to, say, an easement over their neighbor's driveway.  To transfer title you will want a general or special warranty deed.  You may also want to talk with a local residential real property attorney to make sure that the deed is properly drafted and recorded.