In H.M.S. Surprise, Stephen (who has recently been rescued from French imprisonment, where he has been cruelly tortured) and Jack are swimming around the ship. Stephen declares that he would one day like to attempt swimming the Hellespont, and Jack replies, "I am sure you could. If one hero could do it, I am sure another can.""
In The Reverse of the Medal, Jack is convicted of stock fraud and among other things, sentenced to an hour in the pillory. While Stephen hires a group to protect him from thrown objects and heckling from the crowd, it turns out to be unnecessary: pretty much every single sailor currently on shore in Britain shows up in support of Jack, giving him a rousing cheer to boot.
The moment when Stephen's daughter, who had been thought to be autistic, responds to her father - in Irish - for the first time, having been brought out of her shell by Stephen's manservant. And the sequence, immediately thereafter, where she falls in love with the seagoing life.
Stephen's welcome back aboard the Surprise in The Wine-Dark Sea after his epic journey across the Andes following the botched coup in Peru.
The moment when Jack is informed by Stephen that he has finally earned his flag - that is, been promoted to admiral in command of an active squadron. Tear Jerker.
Any moment between Stephen and Christine Wood, but especially the passage in Blue at the Mizzen when he proposes to her. She initially turns him down, but accepts his Breguet repeating watch - his most precious material possession - as a sign that she is seriously considering his proposal, and not too much later accepts him.
In The Nutmeg of Consolation when Stephen finds and rescues his former servant Padeen who has been sent to Australia for breaking and entering. Padeen has been whipped almost to death and is a shell of his former self. When he sees Stephen he says "I knew you would come, doctor dear."
Aubrey and Maturin each have absolute faith in the other's professional abilities. Aubrey is convinced Maturin can bring a man back from the dead so long as the tide hasn't turned, and Maturin believes Aubrey could sail around the world blindfolded and bring it into port on a schedule. Whenever one tries to convince the other that he's only human and things might not go as planned, the other just gives him a wry look, like "Bitch, please, stop fishing for compliments and go save the Empire. Again."
In Post Captain, Stephen and Jack have an escalating argument that ends with Stephen challenging Jack to a duel (officially over Jack calling him a liar and a coward, but ultimately over Diana). The next day, Jack sends him a formal note making it clear that although he stands by the "liar" and "coward" remarks, and they will still be dueling, he only called Stephen a bastard in the heat of the moment and didn't mean it personally. Simultaneously heartwarming and infuriating in the way it underlines the utter senselessness of the whole concept of dueling (the duel never ends up happening, of course).