Probably not. Many of the indigenous languages at that point were only spoken languages and were not written, which is why you aren't finding many, if any, sources in an indigenous language. You should be fine if you focus on one of the European languages (French or Spanish if you are studying an area colonized by the Spanish). In the unlikely event that you need an indigenous language for your area of research, your future advisor will tell you and help you find a way to learn it.

A lot of this depends on your particular subject. For example, if you're doing work that connects to extant groups and potentially their oral histories (a methodology that requires mastery in itself!) you may. However, often the acquisition of those languages happens in the field, precisely because they are not normally taught in an instructional setting. In learning certain southern African languages, that was my only choice--even going to teachers who work in the language wouldn't help, because they were already dealing with native speakers, not second-language speakers. So you may well not be able to acquire a relevant one to you until the need is apparent.

I will say that having such a language, and more importantly a demonstrated ability to work in that language, is a helpful differentiator on the market. But for the era you're discussing, it's not an absolute requirement for the general field. When we did our colonial US search, none of the finalists had a Native American language. Constrast that, if you're curious, to our most recent US West/18th-19th search, where two of the four finalists had research ability in a Native American language. As /u/Mictlantecuhtli points out below, the need is very subject- and geography-dependent.

Even if you don't need it for translating primary sources, I think that it would be a sign of respect and thoughtfulness to learn the language that is or was most prevalent in the main location of your research. It might give you insight into the structure, interests, and investments of the culture you're studying*. Studying the protocols that go along with language in conversational settings with peers and with elders may help you to develop relationships, and thereby gain appropriate, respectful access to oral histories directly from the source in an ethical, non-exploitative manner.

Could you do some research without this knowledge? Absolutely. Could you do more, more interesting, and maybe more groundbreaking research with it? I think so. But you'd probably have to seek out a graduate institution with the programs, training opportunities, and community links to help you on your way.

*For example, as far as I understand it, the Cree language uses different grammatical structures for animate vs. inanimate nouns, but the dividing line does not line up with a Western scientific/biological understanding of what constitutes an animate object.

To move through graduate school you would not need to be fluent in Native American languages. It sounds like it could not hurt, given what you would like to study, but the fact is, most historians of colonial America do not know any Native American languages, much less have fluency. The major reason for this is not necessary euro-centrism, though that doesn't help, but rather that most historians gather their data and evidence from written sources, which, almost by default, are filtered through Europeans and European languages before being recorded (or as they are recorded). This means that there are few truly Native American sources available in that form, so many historians, even of those who look at European-Native American interactions, are not what you would call fluent in any Native American language. Of course, if that will be your main focus, it cannot hurt to learn as much as you can, and also to investigate the Native American cultures you'll study such as they exist today. Generally, a historian of such interactions tries to get as much perspective as possible, by approaching the subject from as many angles as possible. I highly recommend "Facing East From Indian Country" by Daniel Richter for you, as his introduction discusses exactly this issue of recovering Native American perspectives and opinions when the overwhelming majority of information about them and their interactions has been filtered through Europeans and Colonists first. The rest of the book attempts to engage in just such recoveries in a number of instances.