They remain your quarry until the end of your next turn, so you can get 2 attacks in.
First turn:
1. Designate with Hunter's Quarry.
2. Attack. If hit, add 1d6 (or 1d8, forget which).
Second turn:
1. Attack. If hit, add 1d6 (or 1d8).
2. At EOT, designation fades away.