The fist bump between Captain Barnacles and Kwazii in "Giant Spider Crab". It really shows the bond between the Captain and his First Lieutenant.
In The Great Arctic Adventure, when Barnacles calls for help from the rest of the Octonauts, who are on vacation. They quickly stop what they're doing and return to the Octopod. Kwazii immediately takes charge as Acting Captain and successfully leads the team to save his Captain and best friend.
Every time the Vegimals get a scene. Expect this to become a Funny Moment as well.
The ending of The Great Arctic Adventure. Barnacles tells Kwazii' that when he's with his crew, the entire world is his home. The music takes a triumphant leap and it cuts to a picture of the Earth. Aww.