First of all, I assume you meant to ask the question, "Does an electron in a stable (non-transitioning) atomic state experience any movement?" Obviously, an electron that is transitioning between states is moving from one state to the other. But for an electron that is just staying in one stable state in an atom, the question is more interesting. Does it move? The answer could be yes or no depending on how we define motion and what form of the electron we consider to be truly real.