One would assume that quinoa would not pass muster for Passover. After all, wheat and other grains (categorized as chametz) are forbidden during the holiday. Moreover, medieval Ashkenazi rabbis also excluded the consumption of kitniyot -- rice, corn, soy beans, and other foods -- might be confused with the forbidden grains or grown near them and possibly comingled.
Though similar to chametz grains, it turns out that quinoa is not technically a grain (it's actually a member of the "goose foot" family, which includes beets and spinach). In addition, as a new world food, quinoa wasn't even on the radar screen of 15th century rabbis who made determinations about kitniyot and Passover.
So, as long as the quinoa is processed in a in a factory with Kosher for Passover certification, it is generally considered to be part of the Passover diet.