Indeed. And those named Quadrants come from TNG, so it's not something that would have been "outdated" by the "introduction" of Galactic Quadrants.
In ST2:TWoK, it's clear that a Quadrant is a subsection of a Sector: it's curious for the heroes to find the Reliant in the same Sector, but downright mysterious that she would happen to be in the same Quadrant. Also, in TOS, a Quadrant is an area where local conflict involving a star system or two exhaustively defines the "recent history" of the place ("Tribbles", "Armageddon") - the same way a conflict might define Somme or Gettysburg for us, but not Europe or the Western Hemisphere.
The big question then is, does there exist a subsection of some Sector that confusingly happens to be named the Alpha Quadrant?
Timo Saloniemi
In ST2:TWoK, it's clear that a Quadrant is a subsection of a Sector: it's curious for the heroes to find the Reliant in the same Sector, but downright mysterious that she would happen to be in the same Quadrant. Also, in TOS, a Quadrant is an area where local conflict involving a star system or two exhaustively defines the "recent history" of the place ("Tribbles", "Armageddon") - the same way a conflict might define Somme or Gettysburg for us, but not Europe or the Western Hemisphere.
The big question then is, does there exist a subsection of some Sector that confusingly happens to be named the Alpha Quadrant?

Timo Saloniemi