Como dizer "minha boca é um túmulo" em inglês

Minha boca é um túmulo - Mum's the word.

Fig. a pledge not to reveal a secret or to tell about something or someone.

"Mum's the word," said Jane to ease Mary's mind about her secret.
(Thefreedictionary)

I'll take it to my grave.
Aqui tem: como-dizer-a-sete-chaves-em-ingles-t3725.html

You can also say: "your secrets are safe with me"

Once I was watching "How I met your mother", and Lily, one of the characters said:

This flapper, Fort Knox.

Hi, there!
It can be "I will zip my lips".
E.g.: Don't worry about what you told me. I promise I'll zip my lips."
I hope it helped.