Silence is golden
Is a well-known saying.  The Oxford Dictionary of Phrase, Saying and Quotation gives a longer version: Speech is silver, but silence is golden.

If people only spoke when they had something important to say, a deathly hush would descend on the earth.
I haven’t managed to find a derivation of, or source for this.  Any ideas?

Be quiet,
Put a sock in it and
Pipe down are three ways of telling people not to talk.
Pipe up is the opposite, often used when someone says something unexpected.

You could hear a pin drop.
It was so quiet that even the sound of a small object falling to the ground was audible.

I can’t hear myself think.
It is so noisy that it is impossible to concentrate.

Keep mum.
Mummers used to mime: acting without words.  Keep mum means “Don’t say anything!”

Mum’s the word.
I’ll keep that secret.

Shocked into silence perhaps.  In any case it is a much nicer word than the more modern
gob-smacked which implies the violence of being hit in the mouth.

means unable to think of anything to say (due to shock).

Struck dumb
dumbfounded mean more or less the same, but can also mean confounded or nonplussed, which implies confusion, but taken literally, would be having nothing to add.

One man who was afflicted in this way was Zechariah the father of John the Baptist.  His story can be found in St Luke’s Gospel Chapters 1 and 2.

is another word for shocked.

There’s none so deaf as those who don’t want to hear.

Sometimes people don’t listen. Matthew Chapter 13 verse 13 (NIV) ‘This is why I speak to them in parables: “Though seeing, they do not see;
Though hearing they do not hear or understand.”’

Cloth ears.
Rag dolls don’t hear much, do they?

Walls have ears.
Sometimes you can be overheard from another room.

“Actions speak louder than words”
is a proverb.  What you do means more than the promise to do it.