A story's climax is a key turning point in the story and the moment in the story when the conflict is at its peak. Treasure Island has various conflicts going on, but the main conflict is centered around a treasure hunt. Based on this supposed treasure being the goal of so many people involved, it makes sense that finally finding the treasure would be the climax to the story.
Treasure Island almost does this; instead of the climax being Jim or Silver finally finding the treasure, the climax occurs when Silver and his pirates follow the map to the treasure's location only to find that no treasure is to be found. Silver's men begin to turn against him because they believe that he tricked all of them. Fortunately for Silver and Jim Hawkins, they are saved from pirate chaos by Livesey, Gray, and Gunn. This ends the climax of the story and leads into the falling action and conclusion. Jim and Silver are led to a cave where Gunn secretly moved the treasure earlier. Now wealthy men, they load their ship and head home.