Yes, both nose and ear piercings are allowed for women.
What the Qur’an and Sunnah Say
Abdullah ibn Abbas said that the Messenger of Allah, on the day of Eid al-Fitr, went to the women after prayer with Bilal and told them to give charity. They began to throw things [into the sack], each woman throwing in her earrings and necklace.’ (Bukhari #98, #921)
What the Scholars Said
A group of scholars wrote: “There is nothing wrong with women piercing their ears and there is [also] nothing wrong with having the ears of young girls pierced, since this was done in the time of the Messenger of Allah without any rebuke.” (Al-Fatawa al-Hindiyya 5:357)
Imam Ibn Abidin said: “If it [i.e. piercing the nose to wear a nose ring] is considered an adornment for women like in some communities, then its ruling will be similar to the ruling of piercing the ear to wear an earring [i.e. permitted].” (Radd al-Muhtar 6:420)
Shaykh Mustafa Umar