You will complete related achievements by taking part in the Carrot Harvest Contest, Bahamut's Assault event and more means.
As soon as you win the first champion title of the Carrot Harvest Contest, you will complete the “Carrot Harvester” achievement and win certain rewards.
1 win will reward you 1 Carrot Rabbit.
10 wins will reward you 1 Huge Crystal Pack which contains 880 Crystals.
50 wins will reward you 1 Carrot King's Ring (A Level 66 equippable item which can increase base Rage by 30).
In between the Bahamut's Assault event, the player that kills Bahamut will complete the “Bahamut Slayer” achievement and win certain rewards.
1 kill will reward you 1 Young Bahamut.
5 kills will reward you 1 Elder Bahamut.
10 kills will reward you 1 Mini Bahamut.