We don't.

My suspicion is that whoever wrote that thought that the province was named after the people because the administrative unit was created recently. I can't find that claim anywhere else.

Per Katerina Schreiber, the name "Nasca" comes from the Spaniard Don Fransisco Nanasca, an early conquistador who owned the land when the first official surveying took place. This 1580 source is the earliest reference I can find to the use of "Nasca" in an official capacity.

The name was applied to the culture by Max Uhle, who excavated there in the first years of the 20th-century. As was the case for so many other cultures of the Peruvian coast, he simply named the distinctive assemblage of ceramics after the place he found it. Indeed, in early publications there's not always a clear distinction between whether "Nasca pottery" is used to mean "pottery from the Nasca region," "pottery in the Nasca style," "pottery made by the people who used to live in Nasca," and "pottery made by Nasca people."

As for what ancient inhabitants would have called, we don't even know what language they spoke. The Inca and Spanish were so effective in promoting the use of Quechua as the indigenous language that we know very little about what was spoken pre-1400. There are a handful of definitely non-Quechua place names in the Nasca region that hint at the presence of a now-extinct language, but I can find no discussion of this beyond "they're not Quechua."