It is not really the muscle mass, but the muscle type. Because they have been bred for generations for performance (strength), their ratio of fast to slow twitch muscle fibers is at the high end of all breeds that I am aware of. The fast twitch muscle fibers bulk up, which gives them the heavily muscled look (they really don't have more muscle cells - just bigger fast twitch cells). Fast twitch muscle provides great strength, but uses a lot of energy for its contractions...obviously the more energy used, the less endurance the horse will have.
At the opposite extreme are Arabs, which have been bred with a high percentage of slow twitch fibers - smaller muscle cells, a smoother look, and the muscles use far less energy for their contractions...needless to say meaning more endurance.
Just for reference, Thoroughbreds fall right between the two and have about a 50-50 ration of slow to fast twitch muscle.
With this in mind, the "average" Quarterhorse will be a much stronger horse than an Arab, but not even be in the same league when it comes to endurance. There is always variance within a breed of course - you will see some Arabs that suck at endurance, and a few Quarterhorses that are fair at endurance. But all other things being equal, and with good training and conditioning, Quarterhorse are not generally competitive in endurance. However, if you aren't interested in "placing", but just out to have fun, any horse can be used for endurance...you just have to be really careful and monitor their vitals regularly because they aren't conformed for endurance and it can be easy to push them too far...mules will sit down when they have gone far enough - horses generally won't...