DIAGNOSIS The clicking is the internal relay inside the Nest engaging and disengaging. This is normal operation but can be audible if the wall plate is loose or the relay is rapidly cycling.
FIX Press firmly on the sides of the Nest display to ensure it is fully snapped onto the baseplate. A loose connection can amplify the sound. Verify the baseplate is screwed tightly to the wall.
IF THAT DIDN'T WORK Rapid clicking indicates a 'short cycling' command. Check your temperature differential settings. Go to Settings > Thermostat > Temperature Preferences > Heat/Cool Tolerance. A very narrow band (e.g., 0.5°F) can cause frequent relay clicks.
HEADS UP If the click coincides with no system operation, you may have a wiring fault causing the Nest to attempt and abort a call for heat or cooling.