Maybe this doesn't track for people who completed the game. Climbed hundreds of meters and the tracker didn't move.
I've also previously completed the game and this quest didn't track despite climbing up and down Mountain. I noticed the advice about watching the tutorials at the bottom of Mountain - I watched both of them, and then the quest tracked. Thank you!
An achievement popped for me about an hour after finishing the quest. I've played this game once before, for another quest. Oh well.