Here is the dialogue between Peter and David:
Peter: "Mary doesn't know how to speak Japanese"
David: "I don't think she doesn't know Japanese. Her mother is a Japanese."
I am not sure if this is a correct sentence:
1. I don't think she doesn't know Japanese.
It's fine, though I think, "Oh, I think she does know Japanese is more likely.
Actually the dialogue sounds more natural as:
P: Mary can't speak Japanese.
D: I think she can. Her mother is
'A Japanese' is not incorrect, but we would normally use 'Japanes' as an adjective, not a noun', after BE.