As somebody once said, "yes, you can." But why would you want to? My gaited and 1/2-bred gaited horses I have owned over the years ALWAYS demonstrated that they could trot out in the pasture...when they wanted to. And, last summer, my KMH (now 16'3hh--I know, oxymoron height but I think that's because there's a lot of TWH in him) showed me a lot of trotting under saddle. Had to post it. I'm not an expert gaited trainer, but everything I've read about training them is not increase collection in order to build the musculature to carry you and to gait well.
I agree with above. This type of horse is bred for quick/comfortable movement on the trail. They are willing, and will try to barrel race for you. But what if you get really serious about the sport? You WILL have to move on.
Fortunately there is a great appetite now for gaited horses. Find your horse an owner looking for a comfortable ride. I also own a QH, and I have different training plans for him.
BTW, my big guy's head is in the center of my Avatar.