It's true that Doyle never mentioned it, but it's logical that Holmes would want a place he could duck into for a quick meal; or a nap; or to put on a new disguise. Watson does say that Holmes had bolt holes around London, so it would have been foolish for him not to have a membership.

I remember reading about his membership in a non canon work.

If you're a fan look up 'The 7% Solution.' It's available as a novel and a movie; Holmes meets the only doctor capable of curing his cocaine addiction, Sigmund Freud.