No, the order doesn't matter technically, but it may for your preference.
Rise of the Demons brought in cursed dice that has not been supported in future expansions, you may want to stay away from this one.
Quarmegeddon is a great expansion that brings in more variety.
Quest of the Qladiator introduces the 'locked' dice mechanic, you may want this one last.
Quartifacts brings in quests, which is essentially giving you more choices on what to do with your dice, something that helps deepen the game.
My *opinion*, is get Quarmegeddon if you just want more variety, get Quartifacts if you want to deepen the gameplay.
I'll keep this for future reference as well (I have Quarmageddon, but not sure if/when I'll pick up any of the rest).
And is it just me who finds it ironic that an expansion with "Quest" in the title is *not* the one that introduces quests?