Pull the dash panels out and check the connection on the back of the button panel. All the climate and radio buttons are part of one single "button module" instead of having about 100 separate wires back there for everything. Not exactly sure how to take it out on the newer cars, but you probably have to start by pulling off the trim bezel.
Not every button should make a click noise. You might have some other electrical problem. Can you locate the clicking noise more precisely?