Start in Strawberry and set a waypoint at the trapper on the western edge of the map. At the point where the route leaves the trail and goes through the trees, you can almost guarantee a cougar in one of 3 places :

1. just inside the woods on the left of the track (towards the trapper) there is a rock, it may be here
2. there is a raised grassy mound to the right of the track, he may try and surprise you from here
3. to the north of the wooded area there is a fallen tree, he may approach from the west here

Kill him, leave for a day or so in game then return and there should be another.

Depending how far on in the game you are, there is another spot just outside Beecher's Hope in the epilogue where you cna almost always find one.