Upvote:6
You will not be allowed to take this trip without a visa. Although Filipino citizens (and most other nationalities) are allowed to visit Jeju without a visa, the rules for this program say that
You must use a direct international flight or ship to Jeju Island
There is a special program allowing groups of Chinese tourists to transit at other airports on their way to Jeju, but non-Chinese citizens are not eligible for it, nor are individual Chinese tourists not part of a recognized group.
So, you should apply for a visa to South Korea, or change your flight to arrive directly at Jeju without this transfer in Seoul.