Operation Desert Storm was an operation of the US Military in Iraq
I feel a ____ coming on" is a structure that basically expresses the same thing as それ来た
It means you expect something is about to happen that you have no control over.
"I feel a song coming on" -->
Someone's (maybe even you're) about to sing, and it can't be stopped
"I feel a Desert Storm Story coming on" --- he's about to tell a story, and she knows it, and it can't be stopped.
Now, the only reason she's able to expect it is because it's happened before. He probably tells war stories a lot, even when no one wants to hear them.