No, it is not appropriate. Fanfiction should generally adhere to ethical and moral standards, and writing about sexual content in relation to a show in this way is not acceptable. It can also be considered a form of inappropriate and disrespectful use of the original work.
One possible scenario in such a fanfiction could be that Dawson gets hurt during a rescue operation. Maybe she is trapped under some debris while trying to save a victim in a burning building. Her fellow firefighters then have to race against time to get her out safely.
Another theme is the emotional impact on the team. Casey is an important part of the 'Chicago Fire' crew. So when he's hurt, the others might feel guilty, worried, or be in denial. Fanfictions often explore these complex emotions within the team members. They might show how Severide, for instance, tries to be strong for Casey but is secretly very worried. Also, the theme of friendship and support is big. The team comes together to support Casey, whether it's bringing him food during his recovery or just being there to listen to his fears.
One possible fanfiction could be where Matt Casey is injured during a particularly dangerous rescue operation. Maybe he gets trapped under some debris while trying to save a victim. His team members, like Severide, are desperately trying to get him out. The emotional turmoil of his friends and colleagues as they wait for the paramedics could be a major part of the story. And of course, his recovery process could involve some self - reflection and a re - evaluation of his life in the fire department.