When the Franks conquered parts of Gaul, the locals probably referred to themselves as Romans. Is there a specific date where it is shown the locals start to call themselves by the name of their conquerors, as Franks?
edit - typo
No particular date. The change from Gallo-Roman to Frankish culture was a gradual one.
Gaul was fully conquered by Clovis and his descendants by 533 and therefore was under Frankish rule until the Carolingian dynasty ended. The general population was still largely Gallo-Roman so assimilation into Frankish culture took time. By the time Charlemagne inherited the throne of Francia (later France) the culture had pretty much become fully Frankish, although southern France to this day still has a very Occitan culture to it. If you still need a date to say "the population was Fankish by now" then go with 768 when Charlemagne became king of the Franks.