Clicking noise coming from glove box when car is on and off

could be the relay for the ac compressor operating off the thermostat setting

Remove the starter and have it tested at your local parts store.

Blower motor ? Take the glove box , plus lower panel ! When it happens , investigate the noise . You don't say what make , model , year vehicle . Don't know whats back there .

The noise is likely the AC/Heater temperature control blend door moving. Some noise is normal

If you take the glovebox out, you can see the air inlet servomotor on the left side of the blower housing. (SEE DIAGRAM)

This is very likely to be what you are hearing. This is a quite common problem. This servo is electronicall controlled. When the HVAC is turned on, the servo is supposed to go through a "full swing" to test itself. Many times, things (papers, napkins, drinking straws, etc.) from an overflowing glove box get jammed in the recirc-air flap right behind the glove box. when this happens, the servo cannot make a full swing and sometimes it damages the servo, which continues to attempt the full swing and keeps clicking because it is stripping its internal gears trying to make it happen.
You will most likely have to remove whatever it is that is jamming the door and replace the servo.

Its must be your a/c recirculation door motor its located near the right side of the glove box. open your glove box next time it clicks. its in a tough spot to remove but it can be done. you can buy them aftermarket for about $25.00

swirl cage on blower motor is out of balance replace swirl cage hope this helps

If you pull out the glove box you will find the in cabin air filter, you probly want to replace it

the motor that opens the vent is stuck. Put in it fresh air mode. the vent should be on the pass side behind the glove box. Take off the glove box and disconnect the motor and replcae it. Or the cheap way is to not use that mode

sounds like a blend door motor making noise to me.