The 6-foot-4, 300-pound Martin played end in the Chargers' 3-4 scheme but likely will move to tackle in the Dolphins' 4-3 defense.
"I'm a defensive linemen, and in the NFL you're going to have to be interchangeable," Martin said. "My goal is to help the team win. That's ultimately my role."
A native of Jamaica, Martin was a fourth-round draft pick by the Chargers in 2009 and started 27 games the past two years. He missed the final three games in 2012 with a groin injury.
"He is an athletic player that we feel will make a positive impact and bring depth to the defensive side of the ball," Miami general manager Jeff Ireland said.
Martin was recruited by Dolphins center Mike Pouncey, who trains with Martin during the offseason.
"He kind of wears his heart on his sleeve," Martin said. "He's an intense guy. He works hard. I admire his work ethic just from meeting him from the first day. You notice that in five minutes, and he really does love this team and he really was adamant in how of a perfect fit he thought this was for me."