Qualifications
You don't need any particular degree to be a fashion photographer, but if you're serious about it you probably ought to get a BA or BFA in visual arts or photography. All that artistic training might be what sets you apart from the hacks out there. The best fashion photographers are not only hip to contemporary fashion and street style, but they also have a solid knowledge of art history and techniques. Let's face it: Referencing an obscure Max Ernst piece in your shoot for Dior makes you look cool.
Another plus about getting a degree is you'll get expert training (if it's a legit program) in how to use all the cameras and digital editing software you'll need to make your portfolio and website pop. Schools can help you build your portfolios, and they can also set you up with the internships and assistantships you'll need to schmooze your way to the top.
All that school-is-cool stuff aside, you can break into the world of fashion photography with a totally unrelated degree. No one in fashion photography really gives a hoot as long as you have an incredible portfolio. All that's totally necessary to make it as a fashion photographer is buckets of talent, the right contacts, and the luck of the Irish.