Can a graphic designer become a marketing manager?

Yes Indeed you can. You can get into the same field and use your expertise from grass root level to sell it. As a Marketing Manager it will be much easier for you to make the person infront understand from developer point of view. Otherway it will be smoother way of understanding requirement and providing required solution.

Hope this works for you....

Yes strongly agree on that questions  graphic designers can surpass by their limits they can make all what they want in terms creativity because the designers are skilled, talented, resourceful, marketing manager is easy for them.. just like which is hard to learn? .. learn how to drive or how to learn photoshop.... 

Yes If you have great commanding skills then no one will be better than Graphic Designer to become a marketing  manager.  Because Graphic Designer will have all other skills like branding, corporate identity, SEO, Analytics, E-Markeing etc.,