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.
Archive of Our Own (AO3) is a great place to start. It has a large collection of fanfiction, and you can easily search for 'Batman and Green Lantern' with the gay relationship tag. Another option could be FanFiction.net, although the search might be a bit more difficult there as you have to dig through different categories.
It depends on your taste. Some people love the imaginative universe and complex plots of Green Lantern comics. However, if you're not into superhero or sci-fi themes, you might not find them as good.