If I click on the moon item in my inventory and click on 'use' then i get a list of nearby people that i can moon and when i do so, I get the message "You used a moon.". It seemed to work but after I mooned 5 people like this i looked at my quest long and it said I had mooned 0/5.
However, if I click on the player's avatar or their name in the chat log and moon them from that menu, it works. The log says "You mooned Beadle! Energy -1", and it counts towards the quest.