There is a small underground lake in the foggy part of mt. Coronet (the part that connects route 211 and route 216. Feebas will only appear in only 4 different tiles of water that change every day. You will need to surf and try fishing in every single spot to find it in one of those tiles. Feebas only appears 50% of the time on those 4 tiles, so make sure to try each spot at least twice.
On the contrary, Feebas is breedable, and therefore quite common on the Global Trade Station. It may be easier to look for one there.