I don't think you automatically have warts-I believe that with some of the high risk HPV viruses there are no warts-because I have HPV and I have never had warts.
There are over a hundred strains of HPV and only some of them cause genital warts. Some cause other kinds of cell changes (e.g. the kind that can lead to cervical or throat cancer) or some cause plantars or palmars warts (warts that like the feet or hands).