The two characters in the story—an unnamed American man and a woman called "Jig"—are involved romantically. Jig asks the American if soon things will be like they were before and "you'll love me?" He assures her that he does love her "now" but that he is simply worried and preoccupied. He claims that she "know[s] how [he] get[s]." Jig, however, seems more concerned about their being happy and things going back to the way they were "before," while the American is thinking about an "operation" that he clearly seems to want Jig to have.
The point of contention, as you put it, between them is what the American says is "really an awfully simple operation" where "they just let the air in and then its all perfectly natural." Jig seems to be unconvinced by the American's assurances that the couple will be "fine afterward. Just like [they] were before." He claims that "That" is the only thing that "bothers [them]" and makes them "unhappy." We have to do some reading between the lines here, but it seems as though they are discussing the possibility of Jig having an abortion.
This makes sense in the context of the hills that she says look like "white elephants," as well as the title of the story. White elephants were sometimes given as gifts by monarchs in southeast Asia. Because these elephants were considered sacred, they could not be put to work, they would cost a small fortune to feed and house, and receivers could be bankrupted by the cost of caring for these animals. Thus, the "gift" could become a sort of curse, though it was meant to be a blessing. It seems, then, that Jig is thinking of how this baby could be seen as a blessing, that the couple "could have everything," but the American man sees it as a curse to be gotten rid of.