The Great War was a large event that involved four nations. Imperials, Altmer, Nords and Redguards.
All of Tamriel certainly knows about it. Even if the Dragonborn was only a baby when it happened, one is bound to hear about such major events at some point unless s/he was living in a cave.
It even ruins the idea of roleplaying that s/he comes from Cyrodiil. How can you grow up in the Imperial Province and not be aware of the White-Gold Concordat or the destruction of an important faction like the Blades?