HIV spreads only through body fluids. If a mosquito bites an HIV infected individual and then an uninfected individual, will the virus spread to the new individual?
This doesn't take much effort to answer! Googling "HIV mosquito" brings us straight to this blog post, which gives three main reasons (along with more details, links to other resources, etc.):
Only saliva is injected into humans when a mosquito bites and thus HIV positive blood that a mosquito may have previously ingested is never transmitted to other humans.
Unlike mosquito borne diseases, HIV is unable to replicate within the mosquito's gut and therefore is broken down.
HIV circulates in human blood at a far lower level than would be necessary to create a new infection.