The characters refrain explicit discussion of the operation, aside from the man's brief discussion of it, mainly because it is an ongoing discussion, one the details of which they are both well versed.
"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.
"I know you wouldn’t mind it, Jig. It’s really not anything. It’s just to let the air in.’
The girl did not say anything.
"I’ll go with you and I’ll stay with you all the time. They just let the air in and then it’s all perfectly natural."
It is not the first time the discussion has been aired, and it is probable from Jig's comments ("Would you please please please please please please please stop talking?") that she has said it is a conversation she is not interested in pursuing, which is part of the difficulty: She seems finsihed with the conversation, but the man hasn't given up his optimism that he might yet persuade her ("But I don’t want anybody but you.").
The narrator, penned by the autor but different and distinct from the author, doesn't speak of it because the narratorial intent is to, first, set the scene at the railway station, with fertile land on one hand and barren land on the other, and then, second, give the reader entrance into Jig and the man's private conversation; to let the reader eavesdrop so to speak on their conversation; to let you hear just like you would if you were sitting at the table next to the couple.
Sitting at a table next to them, everything you heard in person would be cryptic and unexplained. You'd have to infer from what you saw and heard what the true meaning of the conversation and the true state of the relationship was. The distant objective narrator in"Hills Like White Elephants" provides a portal to the reader to be the eavesdropping occupant of the table next to the couple, thus allowing the reader the same confusion, suspense, curiosity, and intrigue you'd have in person.