Is a vow concluded by saying "yes, that's right" in response to an inner thought of intending to fast for four months?

Such thoughts do not constitute a vow (nadhr). A vow is not established merely by intention; rather, it requires an utterance that indicates commitment. Allah has pardoned this nation for what they talk themselves into, as long as they do not act upon it or speak of it. Therefore, nothing is binding upon the questioner, and she must ignore and disregard these whisperings (waswas).