Is it considered a sound and permissible act to compel one’s wife, relatives, and children to speak only Arabic for the purpose of learning the Qur’an and the Arabic language?

The upbringing of a Muslim Arab in non-Muslim lands and his loss of his mother tongue is a calamity. You must oblige your wife to communicate in Arabic, for it is the language of the Qur’an, and learning it is required. Obliging your wife to communicate in Arabic with her mahram relatives is fine. As for non-mahram relatives, it is preferable not to be expansive in conversation with them for fear of falling into forbidden acts. You must improve your Arabic language so that your children may emulate you.