I am sorry! While the mustang ancenstory of barbs and turks may be correct, my one track mind was only thinking of ONE breed that contributed to the Quarter Horse, primarily. Here are a couple things that got that trivia in my head:

From the American Morgan Horse Association site:

The Quarter horse stallion Joe Bailey was a son of Headlight Morgan. Jubilee King, a famous Morgan, was also registered under the Quarter Horse name of Yellow Jacket. Yet another well-known Quarter Horse sire, Joe Hancock, was sired by Redolent, a son of the Morgan Red Oak. In 1920 the King Ranch of Kingsville, Texas, purchased the two-year-old colt Lucky from the U.S. Government Farm for the purpose of upgrading the stock horses they were breeding. Countless Morgan mares were purchased from farms in Illinois and turned out on the range in the King Ranch remuda but can not be identified as the offspring they produced where shown as being out of “Morgan Mare” in the original AQHA registry books.

And here is a link to a great article on the Morgan influence in the Quarter horse breed.

Justin Morgan's Contribution to the Quarter Horse