It depends on what you mean by "this works".
When you use Shillelagh, you're using a bonus action to buff your weapon.
When you take the attack action with your buffed weapon, you use different stats for attack & damage.
Martial arts grants you a bonus action when you take the attack action using a subset of weapons.
So you can use WIS for one (or two, with Extra Attack) attacks with a club/quarterstaff, but the bonus action attacks must be using unarmed strike, which doesn't benefit from Shillelagh.
Beyond that, there's a little bit of challenges with action economy, e.g. that Shillelagh takes a bonus action to cast and Martial Arts grants you a bonus action. So yeah, it "works" but it's usually not as good as the default bonus Martial Arts grants you for using monk weapons.
Also, YMMV, but as DM I'd require either WIS/1d8 or DEX/monk damage, and not let a high level monk use WIS/1d10, but that interaction is a little less clear.