Complete delivery requests for the Endemic Life Researcher in various areas throughout the game and after her quest in the Elder's Recess (Research Help: Bomb Beetle Capture), wait for her to appear again in the Ancient Forest, where you'll find her in her usual spot along the western end of AREA 5, near the Northeast Camp. Talk to her and she'll tell you that the fruits of your labors - and her research - has led to her finding a new critter for you to hunt. To this end, she'll give you the delivery request Research Help: Phantom Bird, charging you with finding this aforementioned critter, which she provides little information about.
That's what guides are for though, no? This "Phantom Bird" she's after just so happens to be a Downy Crake, a fluffy bird whose location isn't best answered with a "where", but with an "on what". The Downy Crake isn't found on its own, as it's a symbiote that travels on the backs of other, larger monsters. In this case, you can find them hitching rides on Apceros in the Wildspire Waste. Your best bet for finding them is to simply travel to the Southwest Camp in the Wildspire Waste and check the backs of the three Apceros that spawn there. They're gray little puff balls of birdy cuteness that line up on the backs of their Apceros host.
Unfortunately, there's a further complication, as for critters that roost on the backs of loud, armored dinosaurs that often end up as prey for other, larger dinosaurs (well, technically wyverns...), they're rather flighty. Sneaking may not be sufficient cover - to bag these birdies, you're going to want to make use of your Ghillie Mantle. If that's not annoying enough, the Downy Crake won't show up every time you enter the area, so you'll likely need to travel between areas (traveling between camps on the same map is not sufficient).
So... equip a Ghillie Mantle, travel to the Southwest Camp in the Wildspire Waste, head out into the AREA 1, use your Ghillie Mantle, then search the backs of the Apceros there for your birds. If you see them, get close and catch them with capture net. For accomplishing this feat you'll find yourself richer one trophy/achievement.