Is it permissible to give zakat money to the wife's brother, who is suffering from severe injuries and needs expensive surgeries, taking into account his financial situation and without informing him that it is zakat?

If your son-in-law does not possess the amount required for treatment, it is permissible for you to give him your Zakat al-mal. If he possesses part of it, it is permissible for you to give him the remaining amount from Zakat to complete what he needs.