I hate to be "that guy" but it depends on what you mean by quality! I think what you have in mind is something like, does the school educate you well? But the only way you'd know if it educated you well is if you succeed in life, i.e. achieved status. ("No, standardized tests!" -- but how the questions on the test are deemed important is irrevocably intertwined with status to begin with!) So we see that education is a tool for achieving status. It's much more direct if the school just conferred status -- which is what we find.

So yes! Prestige is a great indicator of quality - as long as you define "quality" as "prestigious"! (Which we mostly do anyway.)

If that's not clear (it's hard for me to get ideas out of my head sometimes), I guess another way to say it is: Which causes the other? Because I think "prestige" is an input to quality, whereas you're saying "quality" is an input to "prestige."