If you quit a position can you still list it on your resume?

Depends on the job.. Is this an executive position or a federal job? or is this subway or Burger King?? If it's a high end job you may want to let your new employer know so that way he doesn't do a check and catch you lying.

There can be valid reasons for leaving a job off your resume, but the fact that you quit generally isn't one of them. In fact, you probably want your prospective employer to assume that you quit your former jobs as opposed to being fired from them!