Silent vent fan for home theater closet
One of the products that came out was the "Mac Chimney".
It was a hokey looking plastic "dunce cap" you put atop the Mac.
It worked great. But adding height to the vent area, as the hot air
rose up the chimney, it drew cool air in through the bottom vents. The heat created it's own airflow.
You need air in your closet. A fan will move around the air in
it, but you'll end up with fairly warm air.
My thoughts run to making an intake near the bottom and an
"out" up near the top (in the kick of a stair?).
Otherwise, you're hopes of "no vents, no noise" leave "turn it down a lot" as the best choice.
You can cut a hole in the riser of a stair and put a register type cover over it.