Whoever owns the minimum amount of gold (nisaab), namely 85 grams, and one full hijri year has passed, is obliged to pay zakaah on it, by giving one-quarter of one-tenth (2.5%), whether that is paid from the gold itself or the equivalent after selling it, or from other wealth.
As you do not own any other wealth with which to pay the zakaah, you have to give the zakaah from the gold itself, or sell some of it and pay the zakaah. If someone (your father, brother, husband, etc) volunteers to pay zakaah on your behalf, that is permissible, and he will be rewarded for it.
Shaykh Ibn Baaz (may Allaah have mercy on him) said:
If zakaah is owed by a woman who owns jewellery, and her husband or someone else pays it on her behalf with her permission, there is nothing wrong with that. Zakaah does not have to be paid from the jewellery itself, rather it is sufficient to give its value every time one hijri year passes, according to the value of gold and silver in the market at the time when the year is completed.
From Fataawa Islamiyyah, 2/85
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is zakaah due on gold that is prepared for adornment, if a woman does not have any means of paying zakaah other than selling some of the gold?
The correct scholarly view and the most likely to be correct in my view is that zakaah must be paid on jewellery if it reaches the nisaab (minimum threshold), which is eighty-five grams. If it reaches this amount, zakaah must be paid on it. If she has other wealth and pays from that, there is nothing wrong with it. If her husband or one of her relatives pays it on her behalf, there is nothing wrong with that. If neither of these options is available to her, then she should sell some of it and pay zakaah with that money.
Some people may say: if we do this, soon she will not have any jewellery left.
We say: that is not correct, because if it falls below the nisaab, even by a small amount, no zakaah is required. In that case she will inevitably have some jewellery left.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 18/138
And Allaah knows best.