Can I use 4 x 4 to extend the space from the wall with a tv mount?
That'd work. there are easier ways though.

If your TV mount is already mounted to studs  you could just pack your 4" or  6" of wood behind it and screw that wood to the studs then screw the mount to the wood. there's no need to have the wood extend all the way down to the floor.

I'd use 2x4s drilled half-way through with a spade bit and then screwed through the bottom of the holes to the studs with 4" chip-board or drywall screws