First of all, we would like to stress that Islam is the religion of cleanliness. Generally, it urges Muslims to be neat and take care of their clothes and physical appearance. The Prophet, peace and blessings be upon him, says: "Cleanliness is half of faith." 

As far as Islam is concerned, there is nothing wrong with a woman removing hair inside her nose. Although removing hair in the nostrils is permissible, we would like to stress that hair in that area is created to protect one against illness and irritations of the respiratory tract. 

When a person breathes in through the nose, the air is filtered through natural lines of defense that protect against illness and irritation of the respiratory tract. Nasal hairs (vibrissae) at the opening of the nostrils trap large particles of dust that might otherwise be inhaled. The entire respiratory system is lined with a mucous membrane that secretes mucus to trap smaller particles like pollen or smoke. Hair-like structures called cilia line the mucous membrane and move the particles trapped in the mucus out of the nose. 

Responding to your question and stressing the permissibility of removing hair from the nose,Sheikh Muhammad Iqbal Nadvi, Imam of Calgary Mosque, Canada, and Former Professor at King Saud University, Saudi Arabia, states the following: 

"There is nothing in the Shari`ah which prohibits removing hair from inside the nose. The only prohibition occurs for plucking eyebrows. The reason for this prohibition is that plucking eyebrows is considered an artificial change in the Creation of Allah, and it was always a way of excessive adornment of the pre-Islamic period or jahiliyyah. Thus, there is nothing wrong in removing hair inside the nose." 

http://www.onislam.net/english/ask-the-scholar/morals-and-manners/dress-and-adornment/169161.html?Adornment=

http://300035.dotdothktrade.tech/en/artical/25092/women-removing-hair-inside-nose