Using Search Star Trek Scripts, I find there are 415 uses of "sector", usually to mean a volume of space containing several star systems, in Star Trek up to and including Enterprise.
In TOS (1966-1969) and the TOS movies "sector" is used 41 times, usually to mean a volume of space containing several star systems but sometimes to mean a section of a planetary surface, etc. It was also used to describe a sector of interstellar space 6 times in TAS (1973-74).
There are 19 uses of the word "quadrant" in TOS (1966-1969). It can mean a section of the ship or a section of a planet's surface. But it is mostly used to mean a volume of interstellar space containing several star systems.
This use was most notorious in "The Trouble With Tribbles":
KORAX: Of course, I'd say that Captain Kirk deserves his ship. We like the Enterprise. We, we really do. That sagging old rust bucket is designed like a garbage scow. Half the quadrant knows it. That's why they're learning to speak Klingonese.
"Quadrant" is also used 3 times in TAS (1973-1974) to mean a volume of interstellar space containing several star systems.
TOS and TAS do not specify whether sectors and quadrants are different names for the same thing or different sized volumes of space. Quadrants are "fourths" but they don't have to be "fourths" of sectors. It is possible that TOS quadrants are fourths of TOS sectors. It is possible that TOS sectors are parts of quadrants that are parts of still larger volumes of space.
It was not until the TNG episode "The Price" 13 November 1989, that Star Trek started the present system of dividing the entire galaxy into four vast quadrants many times larger than TOS era quadrants.
TOS didn't invent the use of sectors or quadrants for volumes of interstellar space. James White's "Sector General" series is set in a giant hospital space Station, sector Twelve General Hospital, began in the November 1957 issue of New Worlds magazine.
In the military a sector is
SECTOR— A clearly defined area that a given unit
protects or covers with fire.
SECTOR OF FIRE— A section of terrain designated by
boundaries that is assigned to a unit or to a weapon
to cover by fire.
The military use of "sector" for a geographical area may have suggested the sciecne fiction use of "sector" for a galactographical volume.
Ras Ryman (James D. Brown) published The Quadrant War (1976) "a space opera of war between rival interstellar empires".
This was after TOS, but quadrants of space are mentioned in James Blish A Life For The Stars (1962,1963). In Chapter Four "Schoolroom in the Sky" the history of the space cities mentions quadrants of space two times.
What Was the First Science Fiction Use of Space Quadrants?4
What was the first Science Fiction use of Space Sectors?5