You can start the Hierophant Social Link if you get to know the elderly couple who run a quaint bookstore in the Iwatodai Strip Mall. The couple does bring up the Persimmon Tree at Gekkoukan High School before that can happen. It’s not in the most obvious location, but it’s also not a very large area to search.
In Persona 3 Reload, the Persimmon Tree is located in the school’s “Corridor” area. You can quickly locate this by choosing the Corridor option from the school’s fast travel menu by pressing the L1 button. As an alternative, you can warp there right away using the city map.
You’ll notice an open grassy area from there where some students might be hanging out. Examine the tree by walking to its center in that section. This is the concerned Persimmon Tree. You can then report back to the elderly couple and make your formal introductions.
Make sure you don’t have anything else you want to do that day because you will be spending time with them once you get back. As you progress, you’ll also discover a little bit more about this specific Persimmon Tree, so as you play, anticipate seeing more of this institution staple.