To strengthen the HQ I like to use double lunge lines. The first connects from the bit to your leading hand while the second is fed through a ring on the sursingle and around the HQ to your other hand. You can use the second one to encourage the HQ to step under a bit quicker. It is important in all lunging that you encourage and maintain the quality of the gaits that you would want under saddle - the horse should be straight, slightly bent to the circle, and balanced. [I mention this because I often have seen horses just being run around.] Using double lines it is possible to develop shortening and lengthening the strides, make engaged transitions, and begin to leg yield - all of which strengthen the HQ.
Under saddle, short bits of canter - sometimes just a few strides - just as long as your horse is balanced, no longer. I like to start before going into the corner on the short side and returning to the trot as they come out of the second corner. On a 20 mm circle, start at C for instance, canter a quarter of the circle, trot the rest and begin this sequence again at C. Stop the exercise before your horse gets tired. Gradually lengthen the the canter segment.
Do you have access to riding outside the ring? Are there any gently sloping hills.
Do you use ground poles or cavalletti?
Having a vet look at your horse, as others have mentioned, plus checking saddle fit, cantering in half seat to allow your horse to use its back more easily, and checking with a chiropractor are other areas to check.
It took one of my horses - an OTTB - a very long time to develop the strength and balance for a canter. She was great at galloping, though.
Equestrian art is closely related to the wisdom of life - Alois Podhajsky
Couple of things-as ThreeFigs pointed out, conformation makes it more difficult for some breeds to rebalance toward the haunches. I have ridden some horses that are not only rump high, but with very straight hind legs that makes it difficult for them to bend their hocks and step under their bodies to support more of their weight behind. This is even more difficult for them in the 3 beat canter where one hind leg is on the ground alone during a stride.
As a result, these horses can move kind of like wheelbarrows in the canter with their weight on the forehand and momentum carrying them downward and forward in what is kind of a controlled fall. To try to slow them down often results in a lateral gait which allows them to have two legs on the ground at all times during the stride.
In order to get those hind legs under the body, they need to really lift their abdomens and rotate the pelvis and tuck it under them. They need a lot of core strength to do this, and it is going to take a lot longer for them than it will a horse with uphill conformation and/or a more angled hind leg. But it can be done. Exercises to increase the horse's core strength are important including making sure that the horse is very straight, with his front legs between his back legs at all times, which frees up the spine. A hollow back (which is what happens when you try to raise the horses head prematurely) stiifens the spine will make it impossible for the horse to lift his abdomen and rotate his pelvis.
Keep up the cantering on the longe line--but DO use side reins to straighten the horse. You accomplish nothing if the horse is crooked and his back is hollow.
"Against stupidity the gods themselves contend in vain" ~Friedrich Schiller