The conflict between Captain Smollett and Squire Trelawney is found in chapter nine of Treasure Island by Robert Louis Stevenson. The squire has hired the captain for a treasure-seeking journey, but he did not reveal that information to him. Smollett, then, is unhappy for two reasons.
First, he is upset with Trelawney for not telling him the reason for their journey, something he might not have minded except that the entire crew seemed to know their mission while he did not. He resents being trusted less than the crew and he is upset because treasure hunts are more dangerous than other kinds of journeys. The other key complaint Smollett has is the crew Trelawney hired. He should have been able to hire his own men and likes nothing about those whom Trelawney hired--and we learn later that perhaps his instincts were correct about them.
Captain Smollett's concerns are valid and, in fact, show him to be a prudent captain and a conscientious man. AFter he has aired all his justifiable complaints, he gives this respectful ultimatum:
"I am responsible for the ship's safety and the life of every man Jack aboard of her. I see things going, as I think, not quite right. And I ask you to take certain precautions or let me resign my berth. And that's all."
Doctor Livesey has been a kind of moderator of this conversation, and now he remarks that he was sure Smollett had come into this meeting with much stronger intentions. Smollett says that is true, because he did not think Trelawney would listen to anything he had to say. Trelawney admits that if Livesey had not been present, that is probably exactly what would have happened.
"No more I would," cried the squire. "Had Livesey not been here I should have seen you to the deuce. As it is, I have heard you. I will do as you desire, but I think the worse of you."
"That's as you please, sir," said the captain. "You'll find I do my duty."
And with that he took his leave.
So, the conflict is resolved because of Doctor Livesey's presence. Smollett will be allowed to take whatever precautions he sees fit to protect his ship and everyone on it, and Trelawney will let him do it but is resolved to hold onto his bad opinion of the captain. A third person to mediate and a willingness to listen saves Smollett from quitting and the journey from ending before it even began.