You'd want at least a 250 for a 4H LQ. 

I have a 3 horse 4 Star with a large dressing/tack room and we haul it with the F250. Can hardly tell it's there. 

For any LQ I'd at least go with a 1ton truck (F350, 3500). And fot stabilization I prefer a dually.

We have a F350 dually to pull our 4H LQ. It's an oldie (2002) But my husband babies that truck. 

My daily driver is a F350 SWD, which I pull our 20' stock trailer with. Yes I would be able to pull the LQ, but I feel more secure with the dually.

Just my reference, but I like to have more truck than I really need just in case of an issue.

You will need at least a 1 ton truck (dully is safer) when you move up to a Living Quarters trailer. A 3/4 ton is ok but just depends on how big of a trailer you get, for safty a 1 ton is just better. 

You need to figure that its going to need brakes, tires, and the living quarters ripped out and started over.  It looks like by the size of the fenders that its not a very wide trailer.  Its would not be a bad buy for around $8,000  and figure your going to put a couple thousand into it.