It is the girl who first says that the hills in the distance appear to look like white elephants. After they order drinks, the couple begins to discuss something, without directly referring to it, but it seems as though they are discussing the possibility of her getting an abortion. The man, Jig, says to her "We'll be fine afterwards. Just like we were before." He says, "It's the only thing that's made us unhappy." He encourages her not to be afraid and says that he doesn't want her to do it if she doesn't want to. She seems most concerned that things will be "like they were and [he'll] love [her]." Neither the man nor the woman seems particularly forthcoming; they hardly even seem intimate.
A "white elephant" has become a common term for an unwanted gift that is hard to get rid of. It is really an allusion to a legend referring to the King of Siam (now Thailand), who would give rare white elephants as gifts to people who he did not actually like. On the surface, it seems like a really cool gift, but it was expensive to maintain an elephant. The gift recipient could not get rid of the elephant because it would offend the king, and so they would be financially ruined by the exorbitant cost of the animal's upkeep. In this story, then, the baby that the couple is considering aborting is the white elephant. The girl interprets the hills as looking like white elephants because their symbolism—being a costly and unwanted gift that doesn't feel like a gift but an obligation—matches the way she and the man seem to feel about their baby.