Am I accountable for anything for insisting that my colleague tell me about her disagreement with her friend, knowing that she vowed not to tell me?

You should not insist on knowing what does not concern you, for the Prophet, peace and blessings be upon him, said: "Part of a person's good Islam is his leaving alone what does not concern him." And your colleague's vow not to tell you about the subject of her dispute with her friend is a permissible vow that is not binding, so she is not obliged to do anything by not fulfilling it. Therefore, you are not obliged to tell her anything.