For the past few days Ive been hearing this sound and I have ZERO clue what it could possibly be. I DO know that...

The noise ONLY happens after I drive the car for a little bit and shut the car off.

If anyone has any clue what this could be, please Imk. Any and all help is deeply appreciated!

 

What year is your Z? What engine/trans combo?

Just to confirm, this noise only happens when you shutoff the engine, does it happen immediately after cutting the engine or is there a delay?

My guess with the limited information and listening to the noise in your recording, is that this is intake/engine vacuum related. Possibly a failed or failing check valve for the brake booster, or a failing diaphragm in the brake booster. It could also be a cracked or leaking vacuum hose or connection. If you can reliably re-create this condition, I would pop the hood and do some basic sound checks to pinpoint the location of the noise and go from there.
-Icer