Yes, that is correct. “The quieter you become” is essentially a conditional clause; it’s another way of saying “if you become quieter”. Rewording a sentence can help clarify. “If you become quieter, you can hear more” is correctly punctuated, so this sentence, which says the same thing, is as well.
We put commas after conditions. So if you flip the sentence, “You hear more the more you become quieter”, awkward use of “more” or not, here, there would be no comma since the condition comes second.