After I've reduced it's HP to 1, and put it to sleep, the next time I encounter it, will using the quick ball at this point count as the first turn, or does the first ever encounter count as the first turn?
Yes, it does help. Quick Balls have an increased capture rate on the first turn of an encounter. I doesn't matter if that encounter is programmed to run away.
*Note that this does not work for the Legendary Birds in XY because they flee before you can take any action.