The green rabbit, in this case, is a metaphor for something you want to think of, as opposed to the pink elephant you're trying not to think about. Let's say you're trying to get your mind off of some depressing topic (the pink elephant). Instead of thinking "Don't think about the depressing topic, don't think about the depressing topic" which just makes your mind dwell on it, you pick some other topic that you do want to let your mind dwell on. Specifics will vary wildly between people, but you might decide to think about your next hobby project, or the upcoming movie or sports event or concert you're excited about, or a particularly interesting passage in the book you just read which would reward some deep thought. You'd pick something good, positive, or uplifting; something you know will improve your mental health rather than harm it.

If that's the green rabbit in the metaphor, then at no point would "don't think of a green rabbit" be advice you would want to follow.