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Where does the Argonian person iniate the second fighters guild quest?
It would seem like you are referring to the quest Proving the Deed? which begins in the fourth region of each alliance, aka Eastmarch for the Pact.
If so go to the town of Vulkwasten in Malabal Tor, and approach the fighters guild located there.
You should get a quest from a guild member.
If this doesn't work back track to Marbruk in Greenshade, and find the fighters guild quest there.
I noticed you listed it as the 'second fighters guild quest' if you're actually refering to that you'll find it in Elden Root in the fighter's guild chambers.. though it doesn't seem like that's the, but I figured I'd be thorough.
Barring any issues with the quest giver herself, you should be given Fighters Guild quests at any Fighters Guild hall when you have reached the appropriate rank. From there you are directed where to go to find Sees-All-Colors.