I would definitely expect to hear "The indigenous Brazilian Kayapo people" and "the indigenous American people" but I want to clarify a few things for your sake:
1.) in America we usually don't use the word "indigenous" to refer to our indigenous people (this is purely out of historical development) but how we refer to them is still somewhat controversial. I say Native American specifically because I think it is more accurate than American Indian. But there are plenty of Native Americans who do not like to be called Native American and prefer American Indian. I rarely hear someone refer to them as Indians without a modifier. And under no circumstance should you say "injun." That is offensive these days. For more information, see [this wikipedia]
2. Adjective order in English is not only complex, but almost impossible to memorize outright. First, it's not an absolutely known thing: there are varying lists. Second, it may sound slightly off to an English ear, but you will never ever ever be faulted for it. It is much more useful to work on your English in general and not try to hone in on adjective order. That said, if you're a masochist, [here you go].