How do quests work in League of Legends Dominion matches?

Solution 1:

The League of Legends Devs recently posted a design blog about what went into developing the quests, and why.

An excerpt:

One of the major objectives of the quest system was to prevent the emergence of a metagame that promoted capturing three command points and turtling to hold them for the remainder of the battle. Turtling strategies like this one are contrary to the high-energy combat that we wanted to promote with the new game mode. So, in order to prevent these sorts of stratagems from emerging in League of Legends: Dominion, we instituted quests to allow us to alter the strategic value of each command point on the fly.

A quest triggers at random intervals (I've seen two in one match, before) and will always involve two adjacent capture points, one controlled by your team, and one controlled by the enemy.

Completing a quest, that is, capturing your target point before your enemies capture their target point, gives you a short duration 10% damage buff, and immediately deals 20 damage to the enemy nexus.

Solution 2:

The quest can only be of one type : Capture a specific capture point.

These quest spawn at the same time for both teams. First comes after the 5 minute mark, another one spawns 5 minutes after completion of previous quest.

They always concern two adjacent capture point (both team have and adjacent objective, forcing a team fight between the two points).

When a quest occurs, your team should regroup to go and capture the target. If you accomplish the quest, direct damage is dealt to enemy nexus and your whole team gets a buff (a bit like the nashor buff in standard mode).