Time 7:30 CDT (central daylight time)
Frequency: Triggered - Repetitive
World Name: Jenova
Character Name: Trak Anon
NPC Name: Jalzahn
Monster Name:
Class/Level: 50 WHM
Solo
In-game time (dawn/mid-day)
Area and Coordinates: North Shroud, Hyrstmill
Steps
1: Unequip Relic weapon (by default it goes into Armory Chest)
2: Talk with Jalzahn and click Atma Weapon Enchanement
3: Watch Cutsceen
4: Repeat

Additional information:
PS4 - Atma Item Condition 137% - How I solved the problem was moving the Atma weapon from my Armory Chest to inventory and trying. It doesn't seem to upgrade this weapon when its in the Armory chest.