Whenever I start my car, there will be a brief moment where there will be a clicking noise coming from the center dash.

I took out the climate control unit and tried again and there was no clicking noise so I'm assuming it has to do with that.

Here's a video.

 

There are a few electric motors that control arms to open and close climate control things. Blend door actuators. When they go bad or an arm of one breaks, they click.

 

Thanks rspi for your heater core replacement video, without it I wouldn't have found the actuator, which was right above the heater core enclosure when I took out the glove box.
I took a look at it and it the inside was just a mess. The plastic gears were chewed up and no matter what, the thing couldn't turn at all. Then when I plugged it back in to observe, it was trying to force the door to open even when it was open.