Admiral Quinn is a principle mission contact NPC at Earth Spacedock in the Sol System. He will serve as the main mission contact for new recruits to Starfleet, and will oversee your promotions as you progress.
Admiral Quinn's office is accessible by turbolift, the door to which will be immediately in front of you when you dock at Earth Spacedock, in the Sol system.
Apparently Commander Akira Sulu, his aide, is also in that office. Have you seen him?