To understand what this means, you have to look at the context in which it is found. The words "pure quarter" in this book show up when Ponyboy is talking to Cherry about Soda's horse. Ponyboy tells Cherry that Soda's horse (it didn't belong to Soda, Soda just took care of it) got taken away and sold because it was a valuable horse that was "pure quarter." This means that the horse was a purebreed quarter horse.
Quarter horses are a breed of horse that has been specifically created to be quick and agile. They are typically a little smaller than other horses and are not as fast in a long run. However, they are among the fastest horses over short distances. This is why they are called "quarter" horses. It is because they can beat other horses over short distances like a quarter of a mile.
So, when Ponyboy says the horse was "pure quarter" he means that it was a purebred quarter horse.