Answer
(Fatwa: 1171/755/H=1433)
You are not responsible to pay it. However if you pay it with the permission of your wife then it shall be paid. You have the right to excuse from paying it. In this case it shall be wajib on wife to pay the zakah of her gold. Yes, if the wife is not the owner of that gold rather you are its owner and you have borrowed the gold to wife only for use then in this case it shall be wajib on you to pay its zakah.
Allah (Subhana Wa Ta’ala) knows Best
Darul Ifta,
Darul Uloom Deoband