Is a women's hair profession right according to the Qur'an?

No mention.
But not prepare a naked-headed girls to men eyes.
If for women with hijab, no problems.

In Islam the Women Must cover Her Hair When any man come before her even his male cousine or uncles
And the Hair Profession means Hair cutting aur eyebrow plucking is not Permisible for women when they cross the age of 16

Yes, a woman may work as a hairdresser, so long as your duties are not Haram.
But be prepared, there are a great deal of places that refuse to hire a muslim due to the Hijaab. This is illigal, and if it happens, speak to the police.