The birds don't go to specific things the way the foxes do. As far as I can tell they just lead you towards an interesting thing, and then disappear. Could be a bamboo stand, could be a haiku place, could be a hot spring, whatever.

They're a pain to follow in areas with golden tree leaves, but the Super Focus Listening Extra Slow mode makes it a little easier to spot them if you don't mind having to trudge to do it.