You can refer to the apellido paterno as the paternal surname and the apellido materno as the maternal surname. However, you cannot assume that all or even most English speakers will know what these terms mean without an explanation.
If you want to refer to both surnames together, simply using "last name" is probably not a good default choice, as it's very likely to be taken as referring only to the apellido materno. In some contexts you might be able to get away with "so-and-so's two last names" or "both of so-and-so's last names," though pedants may claim that such phrases are inherently contradictory. "Two surnames," "compound surname," or "double surname" are safer choices in that sense, though unfortunately "surname" is also a word that not all native speakers know.
At times you may have to be more specific or use other workarounds, and depending on the circumstances you might even have to give a brief explanation of Spanish naming customs to your clueless anglophone interlocutor. For example, if your name were Juan Rodriguez Martinez and someone asked for your full name, in some contexts it might be enough to say "My first name is Juan, and my surname is Rodriguez Martinez." But in other contexts, you might want to say something like, "In Spain we use surnames from both parents, and my two surnames are Rodriguez and Martinez. Even though Martinez comes last in my full name, I prefer to be called Mr. Rodriguez."
I'm sorry I can't offer a simpler answer, but Spanish naming conventions are so different from those commonly used in the anglophone world that I don't think there's a single word or phrase that will work in all contexts. And your question doesn't really make it clear what particular context(s) you're most interested in!