A Titanic Effort

With the changes to [currency=1533] in 7.3.5, this quest is no longer available, as you can buy the [item=154879] from him for 300 of the essences instead. However, if you had this quest in your log prior to the 7.3.5 launch, you could still turn it in to get one last "cheap" upgrade. After the patch hit I killed the weekly world boss, which gave me 120 [currency=1533], and was allowed to turn the quest in one last time to get the upgrade. Of course, the quest is no longer available afterwards.

I can confirm that turning in "Supplies" quests (bonus chest from +10k rep with an exalted Legion faction) does not give you any essences. :-( Perhaps it was this way for the other legendary item upgrade quests, but I wasn't aware, so thought it would be good to share the info.

To consolidate info from another wowhead post, here is where you CAN get essences:
- Daily emissary cache (4-10 per cache)
- Daily random heroic dungeon finder (5 daily)
- Antorus boss kills - only receive essences once per boss, regardless of difficulty (i.e. killing the first boss on lfr, normal, and heroic, you still only get essences 1 of those times) (3-6 per boss)
- Mythic + cache (20 essences, regardless of mythic difficulty completed the prior week)
- Weekly PvP quest rewards (i.e. earning 1500 honor in 2s, 3s, and rated BGs) (7 each quest completion)

source: http://www.wowhead.com/news=277232/legendary-upgrade-clarifications-in-antorus?webhook/legendary-upgrade-clarifications-in-antorus

There are two versions of this quest: The initial one, and a repeatable one.
Before you accept or complete the initial one, you are already eligible to get some of the Essences, so that you won't miss out - this is why you see people talking about how they had some before they were on the quest.
However after you complete that first quest, you will have to pick up the quest again or you will get nothing.

Source: I foolishly tried to save questlog space by not taking the quest again and got no Essences. Sent a ticket in which confirmed how the system works.