According to a post on Capcom-Unity, Event Quests will be offered "on an ongoing basis," but require players to achieve a certain hunter rank before taking part. "Gigantic Monster Quests," however are only up for one day, allowing parties of hunters to take on the enormous "Jhen Mohran" monster. Arena Quests allow only two players in a party, fighting a single monster. Some of these arena battles will offer better rewards for quicker completion.
Capcom will announce each month's quest schedule on the first Friday of each month on Capcom-Unity, so set your bookmarks, monster hunters.