Quarter horse trials?

I'm not sure what you mean by QUarter horse trials.
Horse trials (as in three-phase events with dressage, jumping and cross country) are open to all breeds. The way to find out about them is to contact http://www.useventing.com/
United States Eventing Association to get the name of your local club.

For Quarter Horse breed events, such as stakes, shows and futurities, go to the American Quarter Horse Association (AQHA)http://www.aqha.com/ and inquire with them about what's appropriate for your horse at your level.

Hope that helps!