Well, in some fanfictions, he might find it during an adventure in the Forbidden Forest. Maybe he stumbles upon a hidden nest while exploring a new part of the forest that's not often visited.
One challenge could be keeping it hidden. The Hungarian Horntail is a very dangerous dragon, and having its egg at Hogwarts would be against the rules. So Harry has to constantly worry about it being discovered by teachers or other students.
Often, it's portrayed as extremely menacing. Its size and strength are emphasized, with its huge wings and long tail being described in great detail to create a sense of awe and fear. For example, a fanfic might describe the ground shaking as it lands.
The Hungarian Horntail is one of the most fearsome dragons in the Harry Potter series. In fanfiction, it often takes on even more complex and interesting roles. Writers might explore its backstory, like how it came to be in the Forbidden Forest or what its relationship with other magical creatures is. It can be a great antagonist, with its powerful fire - breathing and sharp horns, adding a lot of danger and excitement to the story.
One idea could be to write about a young wizard's first encounter with the Harry Hungarian Horntail. Maybe he's a new dragon keeper at Hogwarts and has to learn how to handle this very dangerous dragon. It could be full of action as he tries to avoid its fiery breath.
In some fanfictions, the Horntail and Harry might have a telepathic connection. This allows Harry to understand the dragon's thoughts and motives, and they work together to uncover secrets in the wizarding world. For example, they might find a hidden chamber that holds a powerful magic item that could change the course of the war against Voldemort.
I'm not sure about a specific 'Harry has a Hungarian Horntail' fanfiction. But you could search on Archive of Our Own. There are tons of Harry Potter fanfics there, and you might find one with this unique concept.
You can start by creating a vivid setting. Maybe the story takes place in a hidden valley near Hogwarts where the Harry Hungarian Horntail has made its lair. Then, introduce a conflict. It could be that the dragon's lair is in the path of a new magical construction project at Hogwarts, and the school has to deal with it without harming the dragon.
One interesting plot twist could be that the Hungarian Horntail forms an unexpected alliance with Harry. Maybe it realizes that Harry is not a true threat but rather a means to escape the control of the Dark forces that are trying to use it for evil purposes.
There are also original characters who are students at Hogwarts but with a special interest in dragons. They might be doing their own research on the Hungarian Horntail and end up getting involved in Harry's task. They could provide Harry with different perspectives on how to approach the dragon, or they might accidentally cause some trouble during the Triwizard Tournament task because of their over - enthusiasm.