The other quests you find out about online such as "King of Fighters" and the mediah furnace storyline don't contain this item as a reward anymore, other quests give it instead, "Glory of the Past" and "The Way Back" quests contain this box as a reward. however these quests are each on their own seperate crossroads quest so it doesn't really matter which crossroad you take if your ultimate goal is to get the black abyss weapon, you can personally look through the questbook yourself at what the quests have on offer for rewards, the first crossroad has 42 quests while the second has 60 quests, so if you only want the weapon take the first one because its faster, otherwise take the second if you want to build up some contribution xp

Есть в России уже всё) с переводом

Can be completed only 1x with each account!
The weapons are like a PRI Weapon, pretty strong, and with a special look.
Good as a startweapon for a new class.

Lahns weapon could be this:
http://bdocodex.com/jp/item/13481/

No las tiren ya que no se pueden recuperar

How can I acquire the black abbyssal weapon for my warrior I've done the level 55 quest step but still nothing happened

it seems that you can now get it for every class instead of just 1 per family
https://bdocodex.com/jp/quest/21401/7/

used to be this quest (only once per family) https://bdocodex.com/jp/quest/4523/17/
now it is for every class https://bdocodex.com/jp/quest/21401/7/

I am guessing this item is no longer in game? The quest series listed in the above comments no longer contains this item, I just completed the series. It is not there. The ability to exchange the item also no longer exists. Another unannounced blunder.

Forget everything you know.
As of last update, you get this once per Mediah main questline -> https://bdocodex.com/jp/quest/21505/14/ (The Way Back)