Wikipedia
Vázquez in Gascon and Spanish, probably under the influence of Basqu and Aquitanian, a language related to old Basque and spoken in Gascony in Antiquity (similarly the Latin /w/ evolved into /v/ in French, Italian and other languages). Other roots describe it as a Galician surname, in use not only in Galicia but all over Spanish-speaking world.
To a lesser extent it also occurs in Portuguese-speaking countries, where Vasco as surname predominates. Vasquez means "[son] of Vasco" and Vasco comes from the pre-Roman latinized name "Velascus" - a name of uncertain origin and meaning, but probably meaning Basque or Iberian. In Galician-Portuguese the pre-Roman name becomes Velascu > Veascu > Vaasco > Vasco.
It is known that in some Spanish-speaking countries, families of non-Iberian ancestry have also adopted this surname. In Colombia and Argentina, there have been instances of "Watzke" and "Watzka" families, of German-Czech descent, Hispanicizing their surnames to "Vasquez". The surname was chosen as being the one most closely resembling their former name; in Italy a similar phenomenon was noted with some "Watzke" changing to "Vasco".
There are also Spanish cognate surnames Velasco or Velázquez).