<p>Sometimes, even if we have a collective noun, we use the plural form of the verb. This is the case with people and police, for example. In your question, I’m pretty sure that people is somehow implied, for the intended meaning is that the Navajo PEOPLE speak an apachean language. Anyway, try to read the sentence aloud with ‘speaks’ instead of ‘speak’ and you will see it sounds extremely awkward…</p>
<p>in this context “the Navajo” is considered as the members of “the Navajo”, not this group
my family owns a helicopter. (the family owns)
my family speak English. (members in this family speak… the “people” is implied)</p>