can people inject botox instead of traditional nose surgery to make the nose higher and prettier?

No. botox is for preventing wrinkles by paralysing the muscles that make your face move, It would have no effect on the position of the nose.

Botox is injected under the skin to make it look younger, even if only for a shorter time than expected or commonly believed. I wouldn't think it could ever replace normal plastic surgery. Now regarding the nose, or any part of the body that one believes needs to be "corrected", I would think should only be done in the more matured, since in younger people the body is still growing and changing. I also believe surgery should be to correct physical deformity, since, How do we know that a nose "not high enough" is  not truly pretty and desired precisely by those who think their own nose is too high up, and want to change it?