Cougars spawn in the western part of the map and also in the Northeast around the waterfall. When you reach the epilogue and Blackwater / Beechers Hope and New Austin opens up there are a few other spawns. I've been attacked right outside Beechers Hope a few times and the most consistent spawn Ive found was near the Trapper at the very western edge of the map.
If you care about the pelt quality, kill the cougar wit a poison arrow. Hit them in the sweet spot and it will wander off and die. If you don't care about the pelt or harvesting resources, dynamite arrows will one shot anything. Fire arrows also work and you get to watch it run around on fire.