They would surely use Imperial names (the Au Ra in the Werlyt questline seem to have, and they are from Garlean occupation territories). Also, she would most likely have the 'usual' Garlean mindset frowning upon gods and eikons and not worship any of the Twelve, else I can't imagine she would be able to have a real career. At the very least there would be no Miqo'te culture. But I do suspect that non-Garleans or citizens that aren't of pure Garlean blood have to face a certain degree of discrimination since the are alike to the 'Savages'.

(I'm talking about Miqo'te, though, not sure about Mithra as we haven't had those in FF XIV so far :P)