Is it possible to have the hpv strain that causes genital warts and never have an outbreak of warts?

Yes: Small genital warts have few symptoms, so many women don't even know they have them, unless the warts are on the outside and they can see or feel them. Warts on the outside (vulva) can itch sometimes. Warts in the vagina and on the cervix are unknown to the women unless the warts are bigger and/or get irritated by sex. A wart can start off as a few infected cells. You see it only when it's bigger.