2003 Pontiac Grand Am Ticking is making me crazy

You need to replace the turn signal switch which is also the lights, wipers, ect its all 1 unit

You might want to check your hazard light button it might not be making contact have you had a stereo installed or took that midddle console out that you had to disconnect the hazard lights

I had the same problem with my '02 Grand Am.
It's most likely your Multi-Function switch, it has grease inside that has some contamination (likely metal filings) that is causing it to constantly or randomly "click".
I was able to take the switch out, clean it and restore functionality. It's pretty easy you'll need some torx screw drivers to remove the switch and to take it apart. Once you've removed the cover on the steering column, disconnect the connectors to the switch, and remove the torx screw on the top. Take note of where the switch sits so you can replace it carefully. There are around 8 screws to take the switch apart, once inside you'll see dark grease has "bridged" some of the connector traces (copper). If you clean the grease from bridging, and if you can put some new clean grease on it should work like new. And yes it is VERY annoying. You can get the dealer to put a new Multi-function switch in for around. $150 likely.