The Origin of the expression "Mum's the word"

This expression dates from about 1700, but mum, meaning "silence," is much older. In 2 Henry VI (1:2) Shakespeare wrote, "Seal up your lips, and give no words but mum."

You can also use "telling you this on the QT"
And the origin of that expression is as follows:

According to Robert Hendrickson, in The Facts on File Encyclopedia of Word and Phrase Origins, the first reference is from a British ballad of 1870, which contained the line “Whatever I tell you is on the QT”. It seems to have been just an abbreviated spelling, using the first and last letters of the word quiet, the mild obfuscation also suggesting a meaning for the expression. The Oxford English Dictionary has a first sighting from 1884: “It will be possible to have one spree on the strict q.t.”. Mr Hendrickson points out that it also occurs in a famous London ditty of 1891, sung by Lottie Collins, which also introduced the famous chorus line “Ta-ra-ra-boom-de-ay”:

A sweet Tuxedo girl you see,
Queen of swell society,
Fond of fun as fun can be,
When it's on the strict Q.T.