Is it obligatory to fast for a month as a vow if I said: "Upon me is a vow to fast for a month if I speak with one of them," and is it permissible to fast it intermittently?

If the vow (nathr) is a vow of obstinacy and anger—and its purpose is to prevent oneself from doing something—then it falls under the ruling of an oath (yamin). In this case, an expiation for an oath (kaffarat yamin) becomes obligatory, which is: feeding ten poor persons, or clothing them, or freeing a slave. If one is unable to do any of these, then fasting for three days.