What is the ruling on swearing by Allah and the Quran falsely, out of fear of exposing a matter that affects another person's life, and is expiation obligatory for this oath?

If you swore falsely out of fear that a man would divorce his wife, there is no blame upon you, Allah willing. This falls under the category of reconciling between people, which is permissible, as the Prophet, peace and blessings be upon him, said: "The liar is not he who reconciles between people, saying good things or conveying good." As for your confession to your cousin about what happened, that is impermissible and constitutes backbiting (namimah), which corrupts relationships. The Prophet, peace and blessings be upon him, said: "A qattat (slanderer/tale-bearer) will not enter Paradise." Therefore, you must seek forgiveness for this action and not repeat it.