Later, while talking with Ron, he mentions that he always thought d'Artagnan might be secretly ambitious.
"After all, he be the best swordfighter on the Trident," admits Ron. "He's been on account fer years, but has never been promoted to an officer. And then here comes ye, in just a few months, already the boatswain!"
It sounds like d'Artagnan has an ulterior motive after all, which doesn't have our best interests in mind. Everyone seemed to think there was no history between Peavey and d'Artagnan, except one crewman; Robert might be working with d'Artagnan, especially if their plan is to capitalize on Peavey's death to advance in rank.
Even if everything Robert said is true, what Ron said makes it look like d'Artagnan plans to use the death to advance beyond you somehow. That might mean making himself higher in rank, or making you lower in rank. It wouldn't be difficult for him to accomplish the latter; all he has to do is expose you as the killer and you'll be executed by the crew.
First, subtly ask about the ship's hierarchy from someone. Find out who takes over a position in the event someone retires from the crew, and specifically,
find out who is next in line to take over Peavey's position.
Afterwards, go talk to d'Artagnan, telling him you're prepared to do it but you want to know his reasoning; otherwise, he's expecting a lot from a person if he can't even tell them that much. Use your charisma to effectively invalidate any excuses he makes for not telling you. If he's completely unwilling to tell you, use conversational tactics to manipulate him into divulging as much information as possible.
Examine his manner of speaking and body language to determine how he feels about you, and if he might be willing to betray you. Thinking of all the time you've spent with him in the past, reflect to yourself about whether or not he seems like the type that would ever go back on his word.
Then talk to Peavey, have some drinks with him (not enough for him to become drunk or disoriented), then use your charisma to subtly ask what his thoughts are on various crew members, nonchalantly gossiping about them, and then bring up d'Artagnan to see what relation they have. Watch his facial expression for his initial expression, and try to determine if he's lying about anything he says afterwards. If he indicates there is something he hasn't or doesn't want to tell you, apply your charisma to assure him it's okay to talk about, and tactfully expose the information he is reluctant to reveal.
I don't know if there's enough time for all these interactions, but it's better to queue too much than too little.