I believe @yoshiki was involved in getting QIIME 1 on BaseSpace, if he has time, maybe he'll be able to respond.
As for QIIME 2 on BaseSpace, that would be pretty awesome! We don't have any immediate plans, but the system is designed from the ground-up to support those kinds of integrations, so maybe in the future. If anyone is particularly interested in developing a BaseSpace app, get in touch!