On Good Eats, a town is being terrorized by a vicious clam (in a parody of Jaws). Alton explains at the town assembly that it's physically impossible for a clam to do that, but the townsfolk will have none of it. Later, it's revealed that the man attempting to catch the clam made the whole thing up to draw in tourists.
The phrase clam up originated in the United States in the early 17th century. It's a great example of the English language's habit of taking inspiration from nature. Just as a clam closes its shell when sensing danger, humans tend to close up and stop talking when they're uncomfortable or in a difficult situation.