Find the Dragon Egg is a main quest in Might and Magic VIII: Day of the Destroyer. It is part of the Form an alliance quest, and is needed to bring the dragons of Garrote Gorge into the alliance. The party can either complete this quest, or the Recover the Dragon Egg quest to get the dragon hunters to join. The party must complete one of the quests, and they cannot complete both.
Deftclaw Redreaver's dragons were fighting to survive against Charles Quixote's dragon hunters. The hunters stole an egg which contained Redreaver's unborn heir, and he dared not attack his enemies until the egg was retrieved. When the Hero of Jadame arrived to convince Redreaver to join the Alliance of Jadame, the dragon responded that he would join if his egg was retrieved.
Enter the fortress and head up the stairs to the four corner guardrooms. The southeastern one has a lever that will open the path to the underground labyrinth beneath the fortress. Head to the southern-most part of the labyrinth and find the secret button that will open the room with the trapped chest containing the egg.
Now, the party can decide whether to bring this egg to Deftclaw Redreaver in the Dragon Cave, or to give it to Charles Quixote in the Dragon Hunter's Camp to complete the Recover the Dragon Egg quest and ally with the dragon hunters.
It is recommended to complete the side quests in Garrote Gorge before retrieving the egg, as the side quests will be much harder when one of the sides is hostile (in fact, many side quests will now be impossible to complete).