When you go to a quest board, you can choose to Join a Quest.
Once in, choose Respond to SOS.
You can then set filters (if you want) and enact a search.
Next you should see a quest category list and you can choose what quest you would like to join.
Then depart on the quest like you would when you Post and you will enter somebody else's quest.
Succeed in that quest and you will be one quest closer to this trophy.
Have fun and good luck!