Welcome to the forum. Give a little more info about your TLX: 6 or 4, Trim level (base, Tech, Advance). How long (seconds, minutes?) is this sound? Front of the car, back? Driver side, other side?
I too hear some "noise" when I shut down and also when I start up. There are a lot more electronics in this vehicle. Many more pumps and motors that are driven by themselves, not by the "fan" belt as in the olden days!
Maybe the parking brake engaging? Do you use brake hold? I know the parking brake engages in my 19 TLX after using brake hold and the putting the car in park. It sounds like it's coming from the back on my car?