I have strong suspicion that many of my readers are of a… shall we say… bookish nature. As in, they are major nerds. So I am counting on that nerd power to help me out.
What I am looking for is instances in Star Trek of good Captain and First Officer interactions. This would include instances where they operated well together, as well as instances where conflict was handled. In fact the latter would be especially nice.
I am hoping that my readers will know of specific episodes where these instances take place. At the moment I don’t quite have the time to watch them all to find out on my own.
*Other instances of good Captain and First Officer interactions outside of Star Trek would also be ok too.
Donal Graeme 
Definitely “Measure of a Man.” Riker is forced to prosecute Data against Picard. They’re in conflict throughout the proceedings, but their structure of captain and first officer remains just as strong despite the situation they’ve found themselves in.
Despite being an example of a conflict, it’s also a fascinating episode full of metaphysical questions. It’s probably my favourite Star Trek episode.
I’d suggest browsing through “memory alpha”, the Star Trek wiki, and just looking through character profiles and episodes. You’re bound to run into more examples there if you need more. 🙂
pretty sure this isn’t what you’re looking for, but… I thought of it immediately.
This made me think of the old Captain/First Officer model for husband/wife that we see around from time to time.
I realize you made no claim that this project had anything to do with intersexual relationships, but if the Captain and First Officer in these episodes are both men it will not be transferable to a man/woman model. If the Captain is a woman and the FO a man even less so. If Captain and FO are both women…ugh.
Pretty much all of the Star Trek series. Each dynamic was different but all were exceptional examples of teamwork. Even when there was friction. Makes me wish that those relationships were possible in real life. But I digress. I’ll get back to you with a few episodes.
TNG episodes:
“Pegasus” Picard is sent to salvage an illegal Federation technology that Riker was involved in, which puts Riker in a bind between loyalty to his captain and duty to Command.
“Measure of a Man” pits Picard and Riker against each other debating in court whether Data is a sentient being.
“Best of Both Worlds” has Picard ‘Borged’ into the enemy general and Riker has to simultaneously outwit and rescue his captain.
A transcript from the pilot “Encounter at Farpoint”:
Picard: One further thing – special favor.
Riker: Anything, sir.
Picard: Using the same strength you showed with Captain De Soto, I would appreciate it if you could keep me from making an ass of myself with children.
Riker: Sir?
Picard: I’m not a family man, Riker. And yet, Starfleet has given me a ship with children aboard.
Riker: Yes, sir.
Picard: And I, uh, I don’t feel comfortable with children. But since a Captain needs an image of geniality, you’re to see that’s what I project.
Hunt for Red October, Ramius & Beodin. Watch the whole thing.
I forgot to thank everyone for their submissions. I appreciate it.