When the Mac first came out, it had no fan. Jobs is fan-phobic it seems. Fine in an office, miserable in a home in summer in western mass and lots of places.

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.

Buy a bathroom vent fan and run the exhaust out a stair or into another deadspace(attic or through a wall up to ceiling space.

You can cut a hole in the riser of a stair and put a register type cover over it.

Agree with previous poster. A bathroom type vent fan would work but you need to know how much air you need to move in cubic feet per minute. Get the cubic feet of your closet and then go to the Broan.com site and check out the ventilator fans. They range from about 50 CFM to 400 cfm. The sound levels range all over the place and are measured in sones. The lower the sone, the quieter the fan. Of course, the quieter it is, the more expensive. Also coule consider a remote in-line ventilator. That removes the noise from the immediate area.