I've actually seen video's off youtube where quails of different species were kept together. (Button, Coturnix and Tibetan in one cage once)
I even asked that question myself on one of the videos with mixed breeds in the enclosure, and the lady told me that she hatched those quails together, meaning that they knew each other from birth.
Though I wouldn't actually suggest having a huge society of different breeds of birds in one enclosure. So maybe buying different breeds of quail eggs, incubating them and letting them all grow up together might actually work. (I haven't tried this yet, so I don't know if it'll work.)
Otherwise, do not mix different breeds of adult quail. Quail are quite territorial, and with 'strangers' in their cage it won't be very nice. 
... Maybe you could get one large cage and put loads of dividers in? xD