Upon exiting the Cavernous Maw you will receive a cutscene and the quest will be completed. If you don't leave through Cavernous Maw you will receive a cutscene when you enter Buburimu Peninsula.
Since the great cataclysm that sundered the land and crimsoned the sky of Abyssea, refugees have been under constant siege from numerous fiends that have emerged in the wake of the disaster. But recently, within each of the accessible abyssean areas, a fiend of tremendous strength has arisen to terrorize the struggling survivors; now it is up to the brave adventurers of Vana'diel to put an end to this scourge and restore hope to the shattered realm of Abyssea. These quests will automatically flag upon checking the Cavernous Maw in the designated zones with a Traverser Stone, or previous banked Visitant Status time, in your inventory.
Note: You must be in range to receive the Notorious Monster's title in order to receive credit for the kill. However, you do not have to be alive when the Notorious Monster dies in order to gain the title and credit.
After slaying the designated monster for the zone you are currently in, exit the zone by using the Cavernous Maw or timing out. Upon clicking the maw outside (when leaving abyssea) or zoning into the zone containing the maw from another zone (when you warped or homepointed), you will enter a cutscene and receive your reward which varies based on number of completed quests. Refer to the chart below for a quick reference to what will be obtained.