Um, I don't think this is quite true.
The quest is a major theme in one subset of fantasy writing. There is a lot of other fantasy out there; there really is more to it than Raymond Feist, Terry Goodkind, Robert Jordan, and David Eddings.
It's also something Diana Wynne Jones parodies beautifully in The Tough Guide to Fantasyland and Dark Lord of Derkholm. With very good reason.
ETA: As for what the protagonist does without a quest, they can fall in love, come of age, usurp a kindgom, inherit, avoid inheriting, solve a magical mystery, steal from the rich, protect others, come to grips with their past, do their job...