After installing a new radio, the turn signal clicks loudly through the door speaker?

Yep, the turn signal sounds come from the radio, the 2012 Chevrolet Silverado does not have a built-in feature to adjust the volume of the turn signal through the speakers. The volume of the turn signal sound is typically fixed and cannot be adjusted by the vehicle's settings or controls. So, it sounds like the radio is defective. Did you change the speakers? If so, this can be the issue due to the OHMs difference. Can you please shoot a quick video with your phone so I can hear the noise? That way I can tell for sure what's going on. You can upload it here with your response.