I know modern philosophers debate whether we have sufficient warrant to conceive of properties and qualities as mutually distinct. The existence of such a debate entails the existence of, ostensibly, mutually distinct conceptions of those terms. So, generally speaking, what do the philosophers who posit a distinction between the two terms regard as distinguishing one from the other?
So, I'm not sure what motivates the position you're talking about. But off the cuff, my response would be:
X's property p is whatever makes the sentence "x is p" true.
A quality is just a particular kind of intrinsic property. Color is a quality. So are texture and temperature.
So, all qualities are properties, but not vice versa. Here's a property that isn't a quality--being a father. It is true of me that i'm a father, but that isn't an intrinsic fact about what I am like, it's true in virtue of relationship I have to something outside of me.