How does the novel Hawaii by James Michener relate to the nine nations concept?
I assume that you are asking about Joel Garreau's idea of the Nine Nations of North America. If so, the best connection that I can see between this book and the concept can be seen in Part III of the book. This part of the book draws a connection between the New England "nation" and Hawai'i.
Garreau sees New England as a place that sees itself as more civilized than anywhere else and which admires "such Yankee virtues as ingenuity and shrewd trading." In Part III of Hawaii, you can see the missionaries, the whalers, and their descendants changing Hawai'i through both of these characteristics.
On the one hand, in the book, you have Abner Hale (in particular) trying to "civilize" Hawaii. He destroys the old gods, tries to prevent hula dancing, and does other things meant to make Hawaii into a piece of New England. At the same time, his children and those of the other missionaries (mixed with whaler blood and some Hawai'ian blood) are succeeding in remaking the islands into a place where Yankee ingenuity and shrewd dealing dominate.
As we go through the book, we see the descendants of Hale and Hoxworth and Whipple and the others dominating Hawai'i because they have used their Yankee ingenuity and shrewd dealing to become powerful.
This is the connection that I see between the book at Garreau's theory. One can argue that the book shows that Hawai'i would belong in New England because of the influence of the missionaries and whalers.