If you want to go up the levels, IMHO the best best is to look for a western performance line. That is, something bred for reining/cowhorse/cutting/ranch. They are bred to sit in the back and lift their shoulders more than the other types.
WR This Cats Smart and Sannman lines have great dressage movement, I’m also a huge fan of Freckles Playboy or Colonel Freckles on the papers as they’re super amateur lines. I’ve heard good things about Kit Kat Sugars and Dual Reys, and Metallic Cats are quite trainable. Many of the reining lines are very trainable, the Colonels Smoking Gun lines dominate (just make sure not to find a spicy individual), I personally think if you find one that traces to Hollywood Jac 86 (ie through Hollywood Dun It etc), they are very kind horses.
Obviously it depends on an individual’s conformation, there are racing/barrel bred and some all-around bred horses who could do it, heck I even knew a halter-bred horse who went to PSG.
Which brings me to my next point: The most important thing is getting a good mind – and that’s relatively common in QHs. If you have that, they’ll do anything for you, and they’ll do it happily and with their whole heart and that makes up way more ground than anyone ever talks about. If you’re good enough to ride them, you won’t believe how far you can get up the levels.
And hey maybe your goal isn’t go go up the levels – all-around horses aren’t my bag, because in my experience they struggle to get above Second, but you might WANT that – they are often gorgeous horses, and very kind, very forgiving, easy horses, and they’ll try their butts off for you at the lower levels and you never have to longe them or worry about them spooking and dumping you at a show.
The other thing is it’s pretty common in my experience for dressage people to think that QHs are worthless. If you want a well-bred, good-minded, well-trained QH in the 4-8yo range that has a solid foundation in western performance of some kind, be expecting to pay $20,000-50,000 easily. There are always weird bargains, you can find someone’s backyard everyday horse for a few thousand, or a horse that got left behind in its training, or you can take the risk on an untrained one, but the good ones don’t come cheap.