The only difference is that the acceleration of the brick on the Moon will be smaller than on the Earth.
Gravity is the effect of the Moon to attract (and be attracted by) objects, with mass, through a FORCE. So, the moon exerts a force on the brick (and viceversa).
This force depends upon the masses of the two objects (and the separation as well) but the Moon has a smaller mass, so the force will be smaller and g on the Moon will be smaller as well!