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I have to say that I haven't heard 'quit the club', so it may be American English. What was the context? I'd guess that it meant the person left or dropped out of something. Let's see what the American Speakers say when they're up and about. ;-)
It is difficult to define it without context, but it is not a common "idiom". It sounds like a phrase that can be used literally of figuratively. In the literal sense, one actually terminates one's membership in a real club. In the figurative sense, it would mean stopping some activity or realtionship.
I used to hang around with cocaine addicts, but I quit the club.