Adding the surname before the given name in fanfic can be a choice based on the context. If you're creating a story set in a particular culture or time where that's the norm, go for it. However, if your story is more fantasy or doesn't adhere to such conventions, you have the flexibility to decide based on what works best for the narrative.
It depends on the style and setting of your fanfic. Sometimes, it can add authenticity, but in others, it might not matter much.