I really like the fanfic 'Lantern's Light in Gotham's Shadows'. In this story, the Green Lantern visits Gotham and ends up collaborating with Batman to take down a powerful and mysterious criminal organization. The story does a good job of showing how their different methods of crime - fighting can work together.
One great 'batman and green lantern fanfiction' is 'The Emerald Knight and the Dark Detective'. It combines their unique skills and personalities really well. The story has them team up against a common enemy that threatens both Gotham and Oa. It's full of action and great character interactions.
There are definitely some. However, the quality can vary a great deal. Some might focus more on action and adventure within the context of their relationship, while others could be more centered around their emotional bond. It really depends on what you're looking for in a fanfiction. If you like deep character exploration, look for stories that seem to have a lot of introspection for both Batman and Green Lantern.
You can find 'batman and green lantern fanfiction' on fanfiction.net. It has a large collection of stories written by fans from all over the world. Just search for the relevant tags and you'll get a bunch of options.