I thought at first it was an Iron Maiden song - but that's 'Be quick or be dead.' The phrase appears to be a biblical quotation and the word 'quick' has an archaic meaning of 'alive.' I don't recall the phrase being used in the series, maybe because Frazer, Jones, Godfrey, Mainwaring and Wilson were not generally known for getting a move on!
"The quick and the dead" is a term more familiar to most because of its incorporation into statements of faith found in most mainline Christian denominations - the Nicene Creed and the Apostle's Creed - but it does come from the Bible. (I had to look that part up: it occurs three times in the King James translation.)
I suspect the men of the Home Guard would by and large be familiar with the reference and what it meant because of Britain's Christian tradition even though the word quick was no longer used in the same way in normal speech.
I'm guessing the title of the Louis L'Amour (western) novel probably refers to being fast on the draw - which could also help someone remain alive.