if you move an electron from a point 'p' to 'q' will its potential energy change?

It depends on the position of pts. P and Q. If the positions P and Q are at different potentials in an electric field then yes the potential energy would change. If there is no electric field then it wouldn't !!

Its like if you place an object at some height in your room (i.e. earth's gravitational field) it has some potential energy and if you place it at a greater height (at higher potential) it has higher potential energy!! But it doesn't change if it is moved at different positions at the floor!!