How do I translate the Book module tree navigation?
I found this that outlines the path to doing this with menus in general:
https://www.drupal.org/node/2182497

After testing it appears that this does NOT work with the Book module menus. All configuration options I have tried thus far have not been successful and I'm guessing the only options left are probably a custom code alter of the book_navigation or a custom book menu all together.

Those steps would included either using the book_navigation preprocess hook:
function template_preprocess_book_navigation
https://api.drupal.org/api/drupal/core%21modules%21book%21book.module/function/template_preprocess_book_navigation/8.2.x

and/or the twig template that renders the menu:
book-navigation.html.twig
https://api.drupal.org/api/drupal/core%21modules%21book%21templates%21book-navigation.html.twig/8.2.x

Hopefully those are a good start.