Murtagh is Jamie's trusted companion. He's also Jamie's godfather, but that doesn't really mean anything considering that Jamie's uncles, Dougal and Colum, might also be two of his worst enemies. Family in the 18th-century Scottish highlands doesn't quite mean the same thing that it does today.
Jamie trusts Murtagh because he's handy in a fight, especially with a gun, and he always has Jamie's back. It's Murtagh who assists Claire in breaking Jamie of Wentworth prison, something she never could have done by herself.
Claire often describes Murtagh early on as "weasel-faced" (3.62), which is ironic given that he's probably the most honest out of all Dougal and Colum's men. We'd even trust him with our own life.