this is always an interesting question. As far as I know, and I'm happy to be proved wrong - there's no nationally recognised professional web design qualification, although there are various design courses you could take.
I've been designing websites commercially for many years now, and in all that time I have never acquired (or needed) any kind of official qualification. Certainly no client has ever asked me if I have such a thing. I taught myself from the beginning, first by learning how to hand-code in HTML, and as the years passed I picked up other skills as and when I needed them.
I've always believed that although you can teach the technical aspects of site construction, site design is an entirely different thing, and to a large extent your success or failure in that field depends on whether or not you have an innate sense of design. Many people don't, and although they can pick up many design do's and don'ts they will probably never become designers in the true sense of the word. I say that in no way to infer any sense of elitism - it's a simple fact of life.
If you feel you have design flair then the answer is to learn about the technical aspects - which you can do by following some of the many online tutorials - and practice. I can't stress the value of hands-on practice enough, to my mind it's the key to becoming adept in this field.
I'm not sure if all that is helpful, or if any of it is, but it's my own experience, gained over quite a long time in the business.