This axiom, regardless of its origins, has been common in post-victorian Britain as an abbreviation of 'to mind your manners' or, more specifically, to say both 'please' (p's) and 'thankyou' (thank-Q).
Thus the phrase 'watch your p's and q's' has been in use to encourage people to speak politely, especially children, who remember such phrases better than just instructions.
Kid 1: You spill my pint?
Kid 2: feck off!
Mum: Oi, watch you're p's and q's!
Kid 2: Sorry. Feck off, please.
by Phil Bool March 22, 2006
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p's = paper as in money

q's = measurement of drugs
by g-unit-gal June 8, 2006
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Shaortening for "Pints and Quarts", which were the two main measurements for drinks. Whereas now we use pints, half pints, shots etc, there used to be Pints and Quarts.

Whehn things got rowdy, as they tend to do in pubs, you'd normally hear something like "Mind your Pints and Quarts" and a warning or some such. this was shortened to "mind your P's and Q's"
by MrJR December 15, 2005
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(Term of phrase used by some and understood by few...)

P's = Priorities
Q's = Qualities

"Mind your P's and Q's."

In lamens terms: Always remember that you have "priorities" in life, and the "qualities" of your character will help you achieve them...
by Skaughty Dawg January 29, 2007
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A packet of M n' Ms in GTA V. Probably means something else comedy-wise that I don't know.
GTA Online player 1: Let's go buy some P's and Q's!
GTA Online player 2: Okay!

*Blows up car*
by ersd123fastman May 27, 2016
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Smile/Face of person holding hands over ears
Symbolizing he/she is not listening
person A: Blah Blah Blah Blah
person B: q>_<p LALALALALA NOT LISTENING
by Nino F. April 23, 2008
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