Is it permissible for a mother to slaughter an 'aqiqah for her son from her own private money without her husband's knowledge, given that the husband would not approve of her selling any of her gold, which she owned prior to the marriage?

The 'Aqiqah is addressed to the father or whoever is responsible for the child's expenses. Therefore, it is not for the mother to perform the 'Aqiqah for her son unless with his father's permission, or if he abstains from performing the 'Aqiqah. It is also permissible for a woman to dispose of her own money without her husband's permission, and it is recommended to inform him as a matter of good companionship.