All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
This gold is considered a Luqatah (a lost and found property), and it is not permissible for you to appropriate it just because you do not know the woman who used to live in the apartment. Rather, you must define that lost and found item by the available means until you find the woman who owns the gold.
We are living at a time and age in which it is not difficult to find that woman, especially that the woman has just moved away from the apartment. So, you can ask and search for her and you will find her. She will also discover that she had lost her gold and return to search for it, so keep the gold in the package in which you found it until you give it back to her.
Zayd ibn Khaalid al-Juhani narrated: “A man came to the Prophet and asked him about the Luqatah? The Prophet said: “Know the particulars of the Luqatah and the string (i.e., know the found object well so that you can question anyone who may claim it later), then announce it for a year. If no one claims it, then dispose of it (i.e., use it or spend it).” [Al-Bukhari and Muslim]
Besides, making a declaration at the police station about lost and found properties make it easy to find their owners.
The Fiqh Encyclopedia reads:
“The person who is making the item public must mention the kind, type and location of the Luqatah, and the date when it is found, especially if the person delayed from making it public; he should also mention its particulars and specifications, because by mentioning its kind, type, particulars and specifications will make the news known and spread among the people and this will lead to finding the owner.
However, the person who makes the item public should not mention all the particulars of the Luqatah, to avoid a liar claiming that it is his and deceive the person and take it instead of its real owner.” [End of quote]
For more benefit on the Luqatah (lost and found item), please refer to Fataawa 133112, 319984, 354181, 88519 and 7604.
Allah knows best.