Muhammad himself was a member of Quraysh. So while he was fighting "Quraysh" as the leaders of Mecca while he was in Medina, it was basically an intra-tribal conflict. The injunction that all future Caliphs must also be from Quraysh is from the hadith (the collected sayings and recollections of the prophet.)

So for instance: http://sunnah.com/bukhari/93/4 and http://sunnah.com/muslim/33/8

In practice, the connection between the Caliphate and Quraysh has been tenuous. The Ottomans were not descendants of Quraysh, for instance.