How we cite our quotes: (Chapter.Paragraph)
Quote #1
"Ye need not be scairt of me," he said softly. "Nor anyone here, so long as I'm with ye." (4.115)
Jamie gives Claire his loyalty almost as soon as he meets her. He has good reason to since he trusted her with his life early on, and she fixed him, cleaning his wounds and setting his dislocated shoulder. If he can trust her with his life, he wants to give her the same security.
Quote #2
For a young man on the run, with unknown enemies, Jamie had been remarkably confiding to a stranger. (7.115)
Jamie trusts Claire with his body (not in that way… at least not at this point) so it's only natural that he goes from that to trusting her with his past. Wounds are both physical and mental, and he also relies on Claire to care for the mental ones.
Quote #3
"Colum Mackenzie, I come to you as kinsman and as ally. I give ye no vow, for my oath is pledged to the name that I bear. […] I give ye my obedience, as kinsman and as laird, and I hold myself bound by your word, so long as my feet rest on the lands of clan Mackenzie." (10.88)
Jamie has to carefully choose his words at the oath-taking due to clan politics. If he pledges an oath to Mackenzie, he becomes one of the clan… which means he might become laird someday. Jamie doesn't want that responsibility, so he manages to earn Colum's trust, but not pledge an oath. Jamie would make a good politician.