Guts is purely fictional. It's a concept or character that exists within the realm of imagination and creative storytelling rather than being based on reality.
The relationship might face some rough patches too. They could argue about things like where to live, how to raise the baby, etc. However, through these arguments, they learn more about each other and eventually find solutions that work for both of them, which in turn, deepens their relationship in the long run.
Perhaps in this fanfic, Griffith has turned into something truly evil and a threat to the world Guts cares about. Guts, being the warrior he is, might feel it's his duty to put an end to Griffith to save others. Also, Guts could be driven by his own sense of justice. If Griffith has wronged innocent people, Guts may decide that killing him is the only way to set things right.
It depends on the specific fanfic. Some fanfics might have Guts killing Griffith as a major plot point, while others could take a different direction and not have that happen at all.