It was to find the SS Botany Bay, which had been sent into deep space centuries earlier.
That small line resolved what would have otherwise been a continuity conflict, as in the original timeline, the Botany Bay was not found until the time of the original series, by Kirk and his crew. Because the ship left Earth before the timeline split, it would have meant that it was still floating out there, and Khan would not yet have been a threat. However, because Starfleet when looking for it, Khan was woken up earlier in this timeline and brought to Earth, starting the events of this movie.
For the aside, "Quadrant" means 2 different things in Star Trek:
- There's the primary 4 quadrants that the galaxy is split into, the Alpha, Beta, Delta, and Gamma quadrants.
- During TOS (and some of TNG), "quadrant" also referred to much smaller sections of the galaxy. For example, Quadrant 904 was mentioned in TOS, and Morgana Quadrant was mentioned in TNG.
Mentioned in the link above, it wasn't until TNG 3x08, The Price, that splitting the galaxy into the 4 primary quadrants was standardized. It was then also used in the Enterprise prequel series and later TOS movies.