Can i run an AC/DC adapter cable behind a wall?
Since your 22v would behind the wall yes this part could be run inside the wall and back out if the wire is listed.
I would get a recessed receptacle and mount the brick behind the monitor , or even a flush monitor and put the brick on the wall behind the monitor.
The only question would be is the wire rated for use inside the wall yes the cable would have to be listed for this, even speaker wire has to be listed to run it inside the walls I have been called several times on new DIY builds where a inspector failed a job because the large clear coated speaker wire was pulled for use in the walls.
There are plastic wire chases that stick to walls(wire mold being 1 brand) These have a snap on cover this would be another choice to consider that would be better than routing the cable in the wall.