If you mean the same time as in the same calendar year, yes. If you mean in 1 location at the same time, no. The drawings are different with moose being first and any-deer later in the summer.

You will need a license beforehand to apply for an any-deer permit but you can put in 1 chance (unless something has changed) for moose. If it were me and you are serious about hunting in Maine I would buy a license before the moose draw.