Oaken

To unlock this achievement you first need to unlock the Pathfinder companion (the dog-like character you meet at the beginning of the game). To find him I believe you need to visit treasure chest locations on the map, which is where companions unlocked.

Once you have Pathfinder, he has a 7 cost Sing card that he adds to The Lady/Kidu's starting deck that will turn any Voiceless monster with less than or equal to 5 HP into a neutral monster. This is what the game considers "converting", which is helpful to know as there is another achievement for converting 10 Voiceless monsters in one playthrough.

Now that you have Pathfinder and the Sing card in you deck, play through to the final boss like normal. Instead of killing the boss' final form like you normally would, use the Sing card on the boss when it is <= 5 HP and pop goes the achievement!