Do you think these games have enough in common to start a serious, competitive club focusing around them? There's definitely enough between Go and Chess, but what about MahJong? I know there is plenty of strategy behind MahJong, having played it, but is it too far removed from the other two?
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I feel there are two major differences:
Mahjong is kinda of like poker or gin rummy in these ways; they more strongly depend on reading your opponent. So, in terms of core properties, I think mahjong is a different category of game to go and chess. Additionally, mahjong is more rules-heavy compared to the rules-light, strategy-rich go and chess.
That said, there's no reason not to play it in a club, as long as there are people who want to play it. Probably the best way to find out if it's suitable for your club is to try making the club, and offering mahjong as an option. If people don't want to play it or don't take it as seriously, then you can move your focus to the other games.
Shogi is an excellent suggestion, being similar to Chess yet having the Asian origins that Go has. If you want to go for the more western games, Draughts or Checkers are closer to Chess than Mah Jong.
The 5 in a row "Go moku" is also a fun, if simple game. It can be played with a standard Go set.
I don't see these as a natural blend with one another, but whether they fit may not really matter. As noted earlier, Mahjong (or dominoes for that matter) is much more luck oriented as opposed to the strategy involved in Chess or Go.
If you're considering starting a club, as with any other group or ahem forum, you need a critical mass of interested participants.
If you're thinking of a club for serious play then your club's focus can be about tournament level play first and the specific game second, in which case you can fit in any game you feel will help you attract participants.
Go and Mahjong can "go" together insofar as they might both attract Asian players. The best place to start such a club is in a physical "Asiatown" in a place like New York City, or perhaps one on line. If you wanted to add "chess" to the mix, I would suggest Chinese chess and/or Japanese shogi in this context.
Otherwise, go and chess are strategic games that both attract "geeks," who probably aren't turned on by the "chanciness" of Mah Jong.
In either case, starting a club is about knowing your audience.