Pokemon Super Mystery Dungeon does indeed feature the fable Mega Evolution, which is a first for the dungeon sub branch series. However, Mega Evolution’s work differently in Pokemon Super Mystery Dungeon than when compared to the mainline RPG Series. Considering there is theoretically no Trainer to Pokemon bond to initiate the Mega Evolution, the developers have found a work around solution to the problem.
Instead of using Pokemon specific Mega Stone’s, Chunsoft has opted for Emera’s stones, more specifically the Awakening Emera’s. Emera’s are items that appear in dungeons and can only be used in the dungeons that you found them from. When you leave a dungeon they will disappear from your bag and Looplet. (Looplets are the item that you attach the Emera’s to).
To Mega Evolve a Pokemon, you simply need to attach an Awakening Emera to the respective Pokemon that has a Mega Evolved form. When you initiate a battle, the Pokemon will naturally mega evolve. For example, to Mega Evolve Absol attach an Awakening Emera to Absols looplet. For Pokemon that have more than one Mega form (like Charizard and Mewtwo), the Mega Evolution version will be selected at random.
For your convenience, below is a list of all Pokemon that can Mega Evolve: