The immediate cause of the fight between Gene and Quackenbush is the fact that Quackenbush questions Gene's reasons for becoming the assistant manager of the crew team. Usually, only disabled kids or kids who want to be manager the next year are willing to be the assistant manager. So Quackenbush is giving Gene a hard time about this.
What finally makes Gene hit him is his use of the word "maimed." I believe that Gene is very sensitive to this word because he feels guilty for having caused Finny to be maimed.