Location: The Hideaway (ship)
Petitioner: Rodrigue, Physicker
Unlocked: Complete Getting to Work
Reward: 40 Sharp Fangs, 40 Steelsilk, 15 Renown
To trigger this side quest you need to complete the quest Getting to Work, which is a sub-quest of the main story quest Bolts from the Blue. Both are fairly simple quests to finish off, and you won't need to complete Bolts from the Blue in its entirety to make Do No Harm appear.
Your petitioner for this quest is Rodrigue, one of the physicians working in the Hideaway's Infirmary. He's been tasked with removing the brand from a Bearer, and he's nervous that he might fail and kill the patient. He agrees to go ahead with it, but Rodrigue needs Errmonea to ease the process. Clive will need to go get some.
Jump to the Hawk's Cry Cliff Obelisk and take a right from the entrance. Before you reach Amber, in the east, you'll come across a group of Riled Raptors and Agitated Hornets. They're just stronger versions of both types of enemies. Wipe out the monsters, then look a little ways east of their location. The Errmonea Leaf you need appears in clusters of blue flowers. Grab the three samples you need and take them back to Rodrigue to receive your reward.