"Hills Like White Elephants" is short, even by the usual standard of short stories. The transitions in it are generally gradual and subtle. For instance, we realize fairly slowly that there is an argument simmering under the surface. One might take the girl's assertion that the man "wouldn't have" seen a white elephant, and his retort, as the first indication, but this could merely be persiflage, too subtle to be regarded as a shift.
The first true shift in the story takes place when the man first mentions the operation:
"It's really an awfully simple operation, Jig," the man said. "It's not really an operation at all."
The girl looked at the ground the table legs rested on.
Until this point, the girl has been speaking more than the man, and seems to have been trying to draw him into conversation. Now, she becomes silent, and he has to speak three times to elicit a reply. These words also introduce the topic which has been the "elephant in the room" between them.
The second and final shift, close to the end of the story, comes when the girl refuses to listen to any more of the man's attempts at persuasion, saying:
Would you please please please please please please please stop talking?
Although he attempts to broach the subject one more time, this is the point at which the story's central topic is essentially shut down, and the two characters address themselves to the practicalities of catching the train and continuing their journey.