A largely unexplored region of space towards which the Enterprise-D was heading in 2365 when it entered the void created by the entity Nagilum.
The Morgana Quadrant is a region of space named in the Next Generation episode Where Silence Has Lease.
The Enterprise-D is on a charting mission towards the area when sensors pick up a patch of total blackness with no measurable properties. Picard moves in for a closer look, and the void closes around the ship.
Inside, the crew meet a copy of their sister ship and then Nagilum, a being that wishes to study death by killing up to half of them. Picard sets the auto-destruct, and Nagilum, having learned what it wanted, releases the ship. The Enterprise then continues on its way to the Morgana Quadrant.