Any horse can do the CTRs....as for endurance it all depends on how competitive you want to be. In the highest levels of the sport it's a LOT of arabians, a decent amount of other breeds do compete though. I know at the CTRs that I've been to there are mostly arabians and morgans BUT....there are also any other breed imaginable...just not as many of them. I've seen some QHs do well, it's all about conditioning. One woman that did really well in Maine this year rides only Shetland ponies...so yes. A quarter horse could do it, especially with proper conditioning. Endurance or CTR really, just know that endurance is more about getting there first where CTR is judged based on conditioning. I think the tendency toward arabs is a lot because they are bred for it, so they don't require as much training and maintenance work for them to be able to complete a race. They generally pace down quickly, and have more of a natural ability. This post is way too long.....yes you can do it!
A horse is the projection of peoples' views about themselves--strong, powerful, beautiful--and it has the capability of giving us escape from our mundane existance.