This term comes from the world of MMORPG gaming, or more specifically World of Warcraft, the most famous of the genre. In these types of games players normally must complete quests to gain experience, money, items, or all three. Normally, a player would have to kill somebody, slaughter groups of monsters, or collect things for a quest giver, however in a fedex quest, all that you have to do to reap the rewards of the quest are to walk around and talk to people and possibly give them items. Players normally love these types of quests because they are essentially free xp and money.
Gamer Nerd 1: Man, you gotta get in on this Legend of Stalvan quest, I've killed like two people and I somehow have like 10000 xp from talking to people! I love fedex quests!
Gamer Nerd 2: CTN, I'm killing Murlocs.