The Noni is Morinda citrifolia, called by locals as bangkoro. The tree is common in Asian and Pacific islands. In the Philippines, I have seen it growing in coastal areas of Bohol, Batangas and Bataan, probably present in a lot more of the coastal provinces. It is said that all parts of the noni is medicinal but recently the fruit was tapped as a dietary supplement for general well-being, hence the juice.
So again what is strange about seeing the noni fruit tree in Quiapo? If you think about it, Quiapo is actually a low lying area near the coast of Manila Bay. It was strategically annexed because of its proximity to Intramuros, the old port town of Manila. The lower part of the Pasig River was probably lined with Morindas in the Spanish era. Maybe we could just assume that the noni in the parking lot is a remnant of the old Pasig River flora. If we put it that way, seeing the noni in Quiapo is not anymore strange...but sad.