Is it permissible for the girls' brother to talk to his sisters' teacher on her personal cell phone about matters concerning his sisters?

A woman's conversation with a man, whether directly or through communication channels, is not inherently forbidden if there is no fear of temptation (fitna) and she adheres to Islamic etiquette. It becomes forbidden if it involves something unlawful or leads to it. If the conversation between the girls' brother is for a legitimate need, observing Islamic guidelines, then there is no objection to it, provided there is no fear of temptation.