To use "either" and "two" is redundant. "Either" means one or the other.
Why is it garotas and not meninas?
Either "garotas" or "meninas" would mean roughly the same in this context. Is there a reason you think "meninas" would be more appropriate?
Technically, 'menina' would refer to a child and 'garota' to an adolescent, but that distinction isn't really used any more (at least in Brazil). I'm not sure, but I think 'menina' is the more polite form in Portugal.
What's wrong with "Does he know one of the girls?"
The sentence clearly shows that are two girls in total. Your suggestion may involve a higher number,
From the tone of the voice it is to understand that it is a question. I have asked before how does one understand that we are dealing with a question?
I think, the word "either" is completely redundant in this English translation