The answer is, as usual, it depends.
Firstly, a pitching moment has to be relative to a defined reference point. Depending on where that point is, any force may produce a pitching moment.
If the reference is taken to be the CG, then obviously weight will not produce any moment as it acts through the cg, but any other force might. The size and sense of the moment then depends on the magnitude of the force and its point of application.
If the reference is a fixed geometrical point (which is sometimes easier to handle) then even weight will produce a notional moment, but when the equations of motion are solved (which must either be about the cg or account for the offset between reference point and cg) the weight effect should be removed.