The character who does not want to leave the island is General Gordon Macarthur.
"Funny, just this minute he didn't want much to get away from the island. To go back to the mainland, back to his little house, back to all the troubles and worries. Through the open window he could hear the waves breaking on the rocks, a little louder now than earlier in the evening. He knew suddenly that he didn't want to leave the island." (Christie, p. 74)
If you mean, what happens to him, how does he end up in the murder plot,
at one point, the other guests find a dazed Macarthur looking out to sea exclaiming, “there is so little time…. I really must insist that no one disturbs me.” He later explains to Vera, “none of us are going to leave the island” and expresses his relief that he won’t have to “carry the burden any longer.”
General Macarthur is found dead, in much the same position that he was found when in was looking out at the sea, except he was hit on the back of the head, otherwise, he is in the same position as when he was first seen.