What I really want is a conflict where all three of them are on opposite sides, and all three feel like the other two agree with each other and are ganging up on the third person. But like I said, I have no plot ideas whatsoever.

(The Earth contingent do something Teyla thinks is stupid; she thinks John and Elizabeth are idiot Earthers and have no idea what Pegasus is really like. It was a decision that was made/undertaken because of/on a mission, and Elizabeth feels like John and Teyla both underminded her authority. John wanted a military response, but was overridden by Teyla and Elizabeth's instincts for diplomacy. Everyone ends up unsatisfied with the outcome, and the decision ends up having been the wrong one anyway -- but it's also clear that any other decision would have been the wrong, one, too.)