Ale used to be served by the tankard which held a quart. You were charged by the angle of your elbow: half way up, you were charged for a pint; all the way up you had drunk a whole quart. Thus you had to be conscious of minding your Ps and Qs.
Somehow I doubt this is the truth.
Beating around the bush: bird hunters used to beat or make noise to cause the birds to fly out so they could be shot.
That's pretty straightforward, even obvious. But it doesn't seem to be compatible with the meaning of the adage, does it? To beat around the bush is to waste time and not come to the point. How were bush-beating hunters wasting time?
I don't believe that one either.
Who has the imagination to put forward other possibilities (more plausible, please) for these lame explanations found in "authoritative" books? By "imagination" I mean YOU think it up instead of going to Google.
(Incidentally, I was long ago taught that minding one's Ps and Qs came from the days when printers hand-set type, and a common "typo" was them putting a p upside down, making it a q. I still like that explanation better. Except one time I realized that an upside down p was a b not a q.)