One good quatermass fanfiction story could be 'The New Adventure of Quatermass' which continues the adventures in a new and exciting setting. It might involve new alien threats and how Quatermass uses his scientific knowledge to deal with them.
Begin by choosing the aspect of Quatermass that interests you the most. Is it his scientific mind? His relationships? Or his role as a hero? Then, build your story around that. If you're interested in his scientific mind, you could write a story where he has to solve a complex scientific problem that no one else can. You'll also need to decide on the genre of your fanfiction. Do you want it to be a horror - themed story, a science - fiction adventure, or something else? Once you've made these decisions, start writing. Don't worry too much about making it perfect at first. Just get your ideas down on paper (or on the screen) and then you can go back and revise it later.
There are classics like 'Bugs Bunny's Easter Special' and 'The Easter Bunny Is Comin' to Town'. They were popular for their charming characters and fun storylines.