You can look up a specific horse's pedigree in the American Quarter Horse Association's online database. As of 2015, these cost between $3 and $15, depending on what information you want. Current AQHA members may be able to access some of them for free.
Try searching AllBreedPedigree.com as well. It provides free simple pedigrees, and there is a subscription-based service that can generate more detailed reports. This information may be less complete or reliable than that of the AQHA because it is user-generated. Horse owners and fanciers add pedigree information for their own horses or for famous sires and dams. This is one way to research general pedigrees if you are not looking for a specific pedigree.
To use these searches, you do need to know the horse's registered name. If you do not have this information but know the breeder, you may be able to submit a description of the horse and the breeder's name to the AQHA and have its researchers help you identify the horse. If it was a racehorse, it should have a lip tattoo that the AQHA can use to identify it. Some quarter horses may also have brands, microchips or other identifying features that you can use to track down their identities.