In Japanese mythology there is the Kojiki and the Nihon Shoki. These two books contain myths but also contain actual history, such as the ties between Japan and the continent of Asia; more specifically Korea and China.

Some of the more famous myths include the ones involving Amaterasu and the creation of the Japanese isles.

What exactly do you mean by "books like the bible and quran"?

Greek mythology has Hesiod's Theogony which goes into the origin of the cosmos and the birth of many gods, monsters, and heroes. The supposed origin of the stories within comes from muses that have knowledge of such things and chose to share knowledge with the author. This is different from Bible/Quran since no deity specifically ordered this story to be told, whereas in Bible/Quran the deity specifically sends a messenger to relay the story.

Hesiod's Works and Days is another book that mostly instructs how to farm and live a stable/moral life. And uses stories about Gods/humans to illustrate the points.

In a way, Homer's Iliad might also fit since it deals with the Trojan War (which has some historical basis) and has significance in explaining human relationships with the gods - specifically that within Greek mythology the Trojan War marked the end of the "sacred era" and afterwards (with a few notable exceptions) the gods chose not to involve themselves as directly with what happens on Earth, but still accept sacrifices and the like.

On the Greek side of things, you'd probably be interested in this thread from a couple of weeks ago.