Can someone please list down the side quest on LoL

There are only 2 sidequests or "events" as they are called officially. These events can only occur in normal and custom games, since they have a too big impact on ranked.

The first one is as you said "The Hunt is On!" (Rengar + Kha'Zix)
The second one is "The battle for Freljord" (Ashe, Lissandra, Sejuani)

There are however some sidequests in the battle tutorial. These include mostly objectives and jungling.

In addition to what the previous poster said, there is still one other sidequest that I know of. I forgot the exact name of the quest, but it revolves around Jinx, Vi, and Caitlyn. Whenever Jinx kills a Vi or Caitlyn, it adds up to his "Officers Jinxed" counter. Every count gives her bonuses iirc, but it's a very small one so it is neglectable. The opposite goes with Vi and Caitlyn, wherein a counter is added whenever you kill a Jinx. This whole side quest is a nod to their lores, in which Jinx is a criminal In Piltover, and Caitlyn and Vi are the police officers there. There are more of this easter-ish eggs in LoL. These things are what makes the game cool to me.