There's no fanfare or quest for this bird, no tasks you need to perform for the Endemic Life Researcher, but there are many similarities between catching this critter and the Downy Crakes you may have hunted for the trophy/achievement "Snuggles for All". This bird - the Bristly Crake - rides on the back of another symbiotic animal, this time a Mosswine, and only those in the Rotten Vale. It's also very flighty, so you'll again need to make use of your Ghillie Mantle.
In the Rotten Vale there are two locations you can find a Mosswine, and hence, the Bristle Crake. From the Southeast Camp, jump off the cliff and turn west to reach the wider platform in AREA 3. After reaching this larger area, immediately turn south to find a hole you can crawl through, beyond which (in AREA 5) you can find a Mosswine.
From the Central Camp, leap off two cliffs to the northwest and northeast, then continue north and turn west and climb up a ledge to reach the chamber (AREA 10) adjacent to the Odogaron's lair (AREA 13). Head to the southern end of this area to find a hole you can crawl through to reach the Grimalkyne cave (AREA 14), then turn west to reach the chamber the Grimalkyne actually occupy. Once inside, turn south and climb up a ledge to reach the walkway overlooking the fishing pond, on which a Mosswine lurks. Travel between areas, as the Bristly Crake may not appear in the Rotten Vale every time, check both Mosswine locations, and repeat as necessary. Once you catch it, you'll pop this trophy/achievement.