The main conflict in the story is less of a conflict between characters, and more the suspense created through Johnsy's declaration that as soon as the last leaf falls off of the vine, she too will fall away, fade, and die. She states ominously, "When the last one falls I must go, too." So, the main conflict is the tension created as we wait, and wait, and wait to see what that leaf is going to do. Other minor conflicts occur as we first find out that Johnsy is sick, and as she continues to get worse. Sue insists that she will get better, and Johnsy is more morbid in her opinion of the matter, so there is conflict between the two on that issue. When Behrmen enters the scene, there is a bit of conflict as he, after Sue tells him of Johny's declaration, expresses his disdain for the silly fancies of women.
I have included a link below to the story itself; I recommend reading it if you haven't already. It is short and entertaining. Good luck!