Would the husband be sinful if he secretly made his mother swear on the Qur’an—without his wife’s knowledge—after his wife accused his mother of stealing her ring, knowing that the mother is accustomed to stealing?

You must deal with problems between your mother and your wife with wisdom, without oppressing either party, and give everyone their due. It is better to separate their living quarters to reduce problems. It is not permissible to accuse a Muslim based on mere suspicion, unless there is strong evidence. If your wife suspects your mother based on evidence, you must mediate between them wisely, and ascertain the truth from your mother if there is proof. If there is no proof, you must administer an oath to your mother, which is permissible. If you believe your wife is truthful and that your mother might have taken the item, compensate your wife kindly and tactfully, urge your mother to repent, and strive to provide sufficiently for her so she is not driven to such actions.