RAW, Shillelagh does not affect the versatile property as the spell wording does not mention that.
RAF, sure go ahead if you like that interpretation as it would then fit with other versatile weapons like the longsword, warhammer, and battleaxe.
Note, that the Monk has a Martial Arts damage die and can affect quarterstaffs and spears as well since they are monk weapons, so that may come up with Monk players once they get to 11th+ level and get a d8 Martial Arts die. Again RAW, the Martial Arts die would not change the versatile property for the quarterstaff or spear to a d10 at 11th+ level. Your game may decide that RAF trumps RAW in this case for Monks. I would be sure to be consistent in both cases of deciding if the RAF options applies to Shillelagh or the Martial Arts die or both for versatile weapons. Note, if you have a Kensai monk, those other versatile weapons like longsword, warhammer, trident, and battleaxe may end up as monk weapons if the Kensai monk takes them as Kensai weapons. So you could conceivably have a 17th+ level Monk using a versatile weapon (quarterstaff, longsword, battleaxe, warhammer, spear, & trident) one-handed at d10 damage and two-handed at d12 damage.
Mike Mearls tweeted that he would allow a quarterstaff's versatile property to be affected by Shillelagh as RAF as a DM, even though it does not as RAW.
technically no, since spell makes no change to that property, but i'd allow it
Jeremy Crawford also tweeted about Martial Arts die not affecting the versatile property RAW.
Martial Arts provides a die that can replace your unarmed strike/monk weapon damage die. It's a replacement, not an increase.