From Middle French and Old French Gascoigne, from Medieval Latin Gasconia, from Vulgar Latin Wasconia, from Latin Vasconia, from Vascones + -ia, from Ancient Greek Οὐασκώνων (Ouaskṓnōn), as used by Strabo in his 1st-century Geographica, Book III. Variously derived from αἴξ (aíx, “goat”) (literally “he-goat people”) or a variant of Ausci (from which Aquitan, Occitan, and perhaps related to Euskara). Cognate to Basque. Compare Gascon.
For sound changes: /v/ → /w/ occurred in the change from Latin to Proto-Romance, while /w/ → /g/ common in (non-Iberian) Romance languages, notably French; compare warranty and guarantee, William and Guillaume. By contrast, /v/ developed into /b/ and /β̞/ in Spanish and Gascon, under influence of Basque, hence the divergence.
Gascogne f