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.
The Chicago Fire started on October 8, 1871. It was said to have begun in a barn. The fire spread rapidly due to strong winds. Many buildings were made of wood at that time, which fueled the fire. It burned for about two days and destroyed a large part of the city.
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.
The Chicago Fire's real story is filled with drama. It all began in a rather ordinary way with a fire starting in a barn. However, the dry conditions of that time, combined with the fact that Chicago had a large number of wooden buildings closely packed together, made it a nightmare. The fire spread so fast that people barely had time to react. It consumed everything in its path, from small houses to large commercial buildings. The city had to be rebuilt from the ashes, and this led to new building regulations and a more modern city plan in the long run.
Another theme is professional growth. Fanfics might show how they learn from each other and improve their skills as firefighters. For example, Casey might teach Dawson a new technique for handling a certain type of fire, and Dawson could share her insights on dealing with victims.