Yes, you can. There is no shar'i problem with that, as long as it does not entail any other form of haram, like non-mahram touching, or, if you are a female, removing of hijab, etc.
Bullies will always bully. If it was not your nose that they wanted to bully about, it would be something else. So your self-confidence should always be high, and this cosmetic surgery could help you, but your spirit must always be strong.
And Allah knows best.