Are these two words interchangeable in most usages?
My specific example is:
It smells bad in this [quarter/quadrant] of the building this morning.
Could both words be used correctly in this sentence; would they have the same meaning?
Quadrant is more specific than quarter as a word. A quarter is 25% of anything, whereas quadrant specifically refers to area. So if you're looking to indicate that exactly 25% of the building is stinky, either word would work.
However while both words are semantically correct in your statement, in terms of common usage they're also both a little odd in the sentence you've used them in. Section, part, area or wing would all be much more common terms for building description.
I can't think of a place where I would use quarter to describe a building.
I would use quadrant to refer to a section of a building, but only when the building has four distinct sections. For example there is a large shopping centre near me that has four coloured sections, I might call them the "red quadrant", "blue quadrant" etc.
I can imagine the word quadrant being used in place of the word sector in laboratories or research facilities but certainly not quarter
A "quarter" is 1/4 of anything. A "quadrant" is 1/4 of a circle, specifically one of the 4 "pie slices" you get by drawing a line that goes through the center of the circle and then another line at right angles to this first line. "Quadrant" is much more specific than "quarter".
You would not normally say "this quadrant of the building ..." Well, not unless the building was shaped like a circle and you indeed wanted to refer to one of the four quadrants.
You would only say "this quarter of the building ..." if you wished to express the idea that 1/4 of the buiding was affected, as opposed to 1/3 or 1/5 or some other fraction.
Several other posters have mentioned that "quarters" can also refer to living space, as in, "the servants' quarters of the mansion", usually used when describing a group residence, like a military barracks or servants' rooms. That's a different definition of the word, but perhaps influenced your question.
The two words 'quarter' and 'quadrant' are not interchangeable in most instances.
If you are thinking of 'part of a general area' (with only a passing connotation of one fourth of the general area), they are both for geographical areas rather than structures (like a building). In that sense they are synonyms (but obviously not when 'quarter' refers to the American coin and 'quadrant' refers to a navigation device).
'Quarter' refers informally to a part of a general area, like a district of a city (where 'quarter is any number of districts of the city not limited to 4).
'Quadrant' is a more technical term referring to the areas of a map which can be split into 4 parts, NW, NE, SE, and SW.
Also, 'quarter' is much more common than 'quadrant'.
I can’t imagine any native speaker saying quadrant of the building. The only architectural use I know of for quadrant is in describing a street curved in a quarter-circle. Even quarter of the building seems unlikely, quarter being used for part of an open area, such as a town. The normal thing to say would be this part of the building.
I could be wrong about this but I always envisioned a quadrant to be a sector of a coordinate space/plane, not necessarily a circle. If you think of the directions on a compass as a coordinate plane rather than a circle, then this understanding of quadrant makes sense. The four sectors of our Milky Way are quadrants. However, looking at some definitions, quadrant clearly has some meanings closely related to circles that I was unaware of.