To help the sound of the stock headunit (base model), I added an EQ that feeds my amp. Now, when I start up or shut down the car, there is a thump from the sub as power is turned off. It didn't exist before the EQ was installed.

I took my 12V power from the antenna green wire to power the EQ and provide amp on signal, since I figured it was a good source of stereo on/off power. Funny thing about the antenna power - it comes on when you start the car, regardless of whether the radio is on or off. It also stays powered for about 3 seconds after you shut the car off.

What can I do to get rid of this thump?

 

First of all disconnect that green wire NOW. its not power its ground. The Z does not have power antenna. (surprise nothing blew up) Radio is on because you completed the circuit with the Amp's power going to the HU too. If you must use an ACC line use the Cig lighter line. The green wire on the bigger wire harness next to the bigger yellow wire is the ACC 12V line. Do not recommend connecting it there. Add a fuse and connected it to the Cig lighter line.

That might get rid of your thump. If not look into buy a delay to power on the sub after the car has started.