No. If they could, then when they were told to full screen (or do so "automatically"), they could draw as convincingly as possible a Qubes-looking desktop and trick users into thinking it's their own desktop, allowing them to phish credentials and whatnot. The viability of this attack is low, but it can be avoided by not allowing full screen.
(Edit: Apparently I was wrong. It is possible to enable fullscreen manually, according to a sibling comment.)