Original: Hagrid is attacked by giants during the mission, and saved by Maxime. Hagrid and Maxime eventually part on the journey home because of Maxime's exasperation with Grawp, Hagrid's half-brother whom he had found and was attempting to bring home with them.
Simplified: Hagrid is also friends with Madame Maxime, it is not known if they were together in a romantic way.