It is said that "The Snows of Kilimanjaro" is Ernest Hemingway's somewhat autobiographical portrait of his life as an author. The best answer to your question is actually found in your question: Harry and Helen quarrel because it is simply what they do. They "drift into it" because that is the nature of their relationship.
Helen is rich and graciously shares her wealth with her husband; however, he seems to resent her for it and even determines to write about a book about the rich and idle class. Though he is now presumably part of that class (by virtue of his marriage), he is disdainful of it. This is one point of contention.
Another cause of the tension between them is Harry's waning love for Helen but also ihs general disdain for all women. Of course this point of view is likely to cause at least one of them (Harry) to be constantly ready to pick an argument.
Harry is dying, and his hopes of getting help and surviving are fading quickly. WHile these circumstances could cause others to temper their bitterness and disappointment so the last moments will not be marred by ugliness, Harry does not care enough about Helen to leave her with anything but their usual animosity.
Finally, Harry has to be frustrated and even disgusted with himself for not treating his injury more quickly; if he had, he could have remained in control of his fate. Since he did not, he now has to face the reality that he is dying because of his own inaction (which is also figuratively true of his emotional relationship with Helen, as well).
Examine the dialogue between the two of them, and it is clear that this is how they are used to dealing with one another. Their disagreements are neither heated nor emotional, despite Harry's use of strong language, which suggests that this is just how they typically talk to one another. Note the following snippet of their conversation:
"Wouldn't you like me to read?" she asked. She was sitting on a canvas chair beside his cot. "There's a breeze coming up.
"Maybe the truck will come."
"I don't give a damn about the truck."
"You give a damn about so many things that I don't."
"Not so many, Harry."
"What about a drink?"
"It's supposed to be bad for you. It said in Black's to avoid all alcohol.
You shouldn't drink."
"Molo!" he shouted.
"You shouldn't," she said. "That's what I mean by giving up. It says it's bad for you. I know it's bad for you."
As you can see, quarreling is a habit with this pair. It is not an ugly, heated kind of quarrelling, but it is the only type of conversation they have left. Early in the story, Harry tells her that he is only talking to make the time pass as he is waiting for the plane to come and rescue him, knowing if the plane does not come he is going to die. Later he makes my point even more clear when he says:
"Talking is the easiest. We quarrel and that makes the time pass."
We should not be upset by this pair's quarreling because they are not upset by it; nevertheless, it does highlight Harry's miserable condition, both emotional and physical. In his mental and physical state, he can do nothing else.