The friendly monster side quest involves locating a number of creatures around the world map in the forested areas. You can find each of them in any order that you prefer, however you cannot find the last Friendly Monster (Yan) until you have found all 8 of the other monsters. Finding Yan before you have found the others will cause it to run away.
When you encounter one of the friendly monsters listed below, it will ask you for an item of some type. You will have to select the “Item” command and ‘use’ the item on the monster.
Located in the Gunitas Basin on the Mist Continent, in the plains near the Village of Dali.
Mu will ask for 1 Ore and the reward is a Potion and 10 AP.
Located in the plains around the Village of Dali and near the Bentini Heights just outside of Treno on the Mist Continent.
Ghost will ask for 1 Ore and the reward is a Hi-Potion and 10 AP.
Located in the Magdeline Forests around the Black Mage Village.
Ladybug will ask for 2 Ore and the reward is an Ether and 20 AP.
Located in the forests near Madain Sari or in the plains near Gizamaluke’s Grotto and Burmecia.
Yeti will ask for 2 Ore and the reward is an Elixir and 20 AP.
Located in the Pualei Plains near the Iifa Tree.
Nymph will ask for 3 Ore and the reward is an Emerald and 30 AP.
Located in the Seaways Canyon, in the forests near Oeilvert on the Forgotten Continent.
Jabberwock will ask for an Emerald and the reward is a Moonstone and 40 AP.
Located in the Mitmakis Ice Field on the Lost Continent (near Esto Gaza).
Feather Circle will ask for a Moonstone and the reward is a Lapiz Lazuli and 30 AP.
Located in the Popos Heights forests which can be reached by climbing up the ladder near Moguta in Gizamaluke’s Grotto shown in the screenshot below. The forest is located just above.
Garuda will ask for a Lapis Lazuli and the reward is a Diamond and 40 AP.
Located on the Vile Island (which is just southwest of the Iifa Tree). Yan will should "Have you met the others?" and run away if you have not found all of the other friendly creatures involved in this side quest.
Yan will ask for a Diamond and the reward is a Rosetta Ring and 50 AP.
Finding Yan completes this side quest and also affects the final battle against Ozmania by making it vulnerable to Shadow based damage.