Let his fire go out!

(1) Is this expression an idiom & what does it mean?
‘Let the fire go out’.
(2) what is the meaning of ’ quarter day’?
‘He had never before sent it in till quarter day’.

I would not say that “let the fire go out” has the status of an idiom or expression (at least, not so far as I know). However, it is a valid sequence of words that can be used literally, or figuratively with “fire” as a metaphor for love, enthusiasm, ambition, etc.

For “quarter day” see e.g. collinsdictionary.com/dictio … uarter-day