constraining push button input using TimeQuest

Your input isn't constrained until you put a set_input_delay constraint on it(or set_max/min_delay). Until you've constrained it, set_false_path doesn't do anything to the design. So no, you don't have to.

Now, there is an Unconstrained Paths report that tells you what ports don't have constraints. Your push-button would show up here, and so sometimes people put a set_false_path assignment on it just to get it off this report. It has no effect on the actual design, but can help the design process(e.g. managing which ports still need constraints).