Some . . . . First of all, I doubt it is actually "Drim..." - more likely "Drum..." which means ridge or long hill in Irish. There are many townlands that start with "Drum" - I don't know of any that start with "Drim." Also, it could be a town, townland or civil parish - they were not consistant on what they gave as an address.
http://www.thecore.com/seanruad/cgi-bin/iresrchThe above is a link to IreAtlas showing all the townlands in Tyrone that begin with "Drum." If you can't get to it via this, just go into IreAtlas and put in Drum in the townland space and set it for "Beginning of word."
Note there is a Drumquin Town in both Longfield East and Longfield West, and a Drumquin townland in Longfield West. One of those might be a good bet. There is also a Drumreagh and other townlands that might be confused with the sound of Drimquay.
If everything else fails, are you familiar with the Ulster Historical Foundation and possibly a member? They have a database that I believe has sub-denominational names in it. These are small parts of townlands. For example, some of my ancestors came from Drumcrow townland in Killyman. A sub-denominational name for part of that townland is Derrybury Drumcrow. I am not a member and have not had a look at the database as yet (so I could be wrong), but I have wondered about it. Having access to sub-denominational names could make Irish research much easier.
Good luck!
Terry