If you hit someone hard around the ear, it swells, and gets thicker.

A thick ear = a swollen ear. This is a threat that you will "box their ears", ie. physically hit them on the side of the head. A similar threat is "I'll give you a thick lip." It's more than just a tongue-lashing.

to present a thick ear means to intentionally not to hear something that is being said. It can also mean to intentionally ignore something which is being said. example, my boss presented a thick ear when I asked for a promotion. hope it helps. :)