Competitive quests are those for which you have to compete with others like Snowpang Island, Death's Hold, and Illusion Isle maybe more but only tested with those 3 and they all worked.
For Death's Hold you have to do the first co-op event by picking up the swords after that the second quest will be a competitive one which you have to fail by not reaching the minimum kills (no one on the Island should reach the minimum had someone get once 5 kills and even tho I had 0 it still counted as completed so beware).
On Illusion Island you can literally just sit afk the whole time if there is no one else and it will fail and count for the achievement.
Snowpang also works but you need to pick a time with not so many people going there otherwise they will finish the quest and not count towards the achievement.
Pvp always has been dead in this game at least on my server - Lazenith EUC I never had problems finding empty pvp islands.
If there is someone else who wants to actually farm the Island I usually just come back another time and let them do the event cuz many many times there is not even a single person trying to complete it which sucks because for some like Deaths Hold you actually need at least some others to actually start the event that you have to lose for this achievement.
This is easily the best option now. All other suggestions (Death's Hold, Illusion) are outdated since they both require a min amount of people to start now which will not be the case most of the time.