It is possible that some car models have the name or branding Quarterback, sometimes there will be a limited edition version of a car which will have a special name, such as 'Ford Focus Quarterback' (this car does not exist as far as I know!). There will usually be some form of branding involved with a team or player to help sell the car.

I'm not familiar with any kind of car called a "quarterback." A "hatchback" is a kind of car in which the rear window and part of the back are hinged, and pivot up together, to give better access to cargo space. The word "hatchback" is slightly similar to "halfback," which is a team position in football. http://dev.hatchheaven.com/wp-content/uploads/2011/02/datsun_cherry_FII_hatchback_77.jpg