This is probably the best reference for the concept you're looking for:
Your basic three-note chord contains two intervals: from the root to the third note above it in the scale, and from the root to the fifth above it. Both intervals have a quality, as do all of the intervals in the 12-note western system. If you go through the major scale and construct each of the 1-3-5 chords, one for each note in the scale (the so-called harmonic structure of the scale), you find out a couple important facts.
The first is that all seven chords have a "perfect" fifth interval except for the seven chord. That one has a diminished fifth interval, but the seven chord is usually left out of songs or modified, so in practice the fifth is almost always perfect.
That leaves the third interval, which is much more interesting. In three of the chords, it's major and the other four are minor. So there's more going on here. Because the fifth isn't very interesting, the quality of the chord as a whole is taken from that interesting third interval.
Why the explanation? Because your question about the "single" notes quality is one and the same with the chord's quality, namely the chord quality is just a shorthand reference the the third's quality. So they're the same notion.
The only nit I'd pick with your question is that there isn't any idea of quality without the comparison of two notes to one another, so a single note such as A has no quality. The third note of a scale, however, does have a quality because it is compared to the root note of the scale. For the twelve possible notes in the western system, the intervals (with their qualities) go:
Minor second, major second, minor third, major third, perfect fourth, augmented fourth/diminished fifth, perfect fifth, minor sixth, major sixth, minor seventh, major seventh
Technically, any perfect can then be further augmented or diminished (making it no longer perfect), which means I missed some overlapping possibilities in that list, such as the major third/diminished fourth overlap. I left those out to keep the list legible.
Likewise minors can be further diminished and majors can be further augmented, but not vice-versa.
Intervals are really the fundamental unit of music and emotion, so it's a good question and it doesn't hurt to spend some time getting to know the language of them so you have a way of communicating about them aside from just playing them.
If you really want to bend your mind, check out just intonation on wikipedia and learn why the western system isn't perfect (the Pythagorean comma, for example) and why there exist other systems than the 12-note system.