There are a total of three trolls throughout the game. For this achievement (I'd recommend doing this on Dark difficulty, just to avoid any frustration), you are going to need to not kill any of them. Completely ignoring them will not net you the achievement.
There is one troll in this chapter. First off, you can't just ignore him and get the achievement. You must participate in his side quest to count for the achievement. You can always find it by the ruined bridge, but just to make it easier, take the quest "Troll Trouble" on the noticeboard outside of the inn. Upon going to the troll, you will fight it a little bit, but the fight will stop when the troll's health is about halfway down. He will tell you that if you help him find out what happened to his wife, then you will be able to pass. Agree to this, and it will start a subplot about finding the wife. Talk with Zoltan and Sile at the inn in Flotsam, and you will then be tasked with searching Sendler's house (close to where Cedric is located). He will tell you he was sold the head of the she-troll to him by a bandit named Dmitri. He will tell you that the bandit is located at the cemetery which is right by his house. Kill the bandits and then return to the troll telling him about his wife's fate. He will let you through after that.
Towards the end of the chapter (regardless of whose side you're on), you will encounter two trolls. The first is a she-troll being attacked by some mercenaries, so kill them to help her. Follow the path a little further and you will meet her husband, who is looking for her. Obviously, don't attack the husband or select the dialogue that prompts a fight. Agree to help him and go back to the she-troll to tell her to go back to her husband. Convince her to go back to her husband to complete the subplot.
The achievement will unlock after you help the last troll in Chapter II.