You will need to break the latent on the Gun of Trials before completing the rest of the Quest.
Breaking the Latent
With the Gun of Trials equipped in your Main hand, fight experience-yielding mobs ("Incredibly Easy Prey" or higher) and perform Weapon Skills or close Skillchains to accumulate 300 "Trial Points".
You know you have enough points when the latent breaks. You will know it is done when the weapon's elemental resistances disappear from your active resistances (colored dots below the gear grid, but above the selected item description). The item's description text will never change.
Take the The Boyahda Tree Home Point or The Boyahda Tree Unity Warp, this will put you on Map 2, run to (G-10) and go south into a tunnel leading to Map 3. Inside, continue south and examine ??? at (J-8) to spawn the NM.
Inside the Tree, follow the path until it opens into a large room; this should be around (I-8). Travel northwest to (G-7) and fall over the ledge. Continue northwest to (C-4) and enter the tunnel there. Follow it as it curves to the south, and then travel south to the tunnel at (A-6).
When you exit the tunnel, you will be in a large area with several waterfalls leading down and a large tree in the center. This area is populated with Korrigan and Elder Goobbue. Travel south to (G-10) tunnel, just inside is the ??? to spawn the NM.