After you win his trust, Komuxio asks you to investigate sightings of strangers in the Shroud.
This quest is available if you have completed the quest Forest Friend.
Starting the Quest
Talk to Komuxio in Little Solace in the East Shroud to begin the quest.
Investigate the Suspicious Presence
There are four quest markers on the map that you can check on in any order. For example, go east and talk to Nathaxio, then go west and talk to Pelixia, then go west and talk to Monne, then enter the Hawthorne Hut and talk to Victor.
Speak with Komuxio
Talk to Komuxio and he will mark your map with the area to investigate for Imperial soldiers.
Search for Clues
Go west to the small area marked on the map (it's east of the Hawthorne Hut aetheryte). You will find a purple circle with a Suspicious Box in it. Enter the purple circle and you will be attacked. Kill the Signifer, then the Eques and Secutor, and finally the Laquearius. Then check on the Suspicious Box.
Deliver the Quartermaster's Log
Go back to Komuxio and talk to him to hand over the Quartermaster's Log.
Next Main Scenario Quest
After you complete the quest, you will need to be at least level 22 to proceed to the next quest. After you reach level 22, talk to Komuxio to accept the next quest: Brotherly Love. There will be a cutscene.