A headrest moving device which includes a press member, a moving unit that is moved horizontally by the press member, a return spring that functions to return the moving unit to an original position, a screw, a block that is provided with threads in the inner circumferential surface thereof and is engaged with the screw through a screw type engagement, a locking unit that selectively releases the block from a locked state in response to a movement of the moving unit, and a frame that is installed on a stay rod and supports the moving unit, in which either one of the screw and the block can be rotated so that the device can control the position of a headrest steplessly in forward and backward directions.