The story doesn't state what time of day it starts. If the question is asking you to literally guess a time of day, then go ahead and pick any time of day and defend it. It's a guess, and most readers are capable of making a guess; however, I do think the opening paragraph steers the guess toward early morning.
He came into the room to shut the windows while we were still in bed and I saw he looked ill. He was shivering, his face was white, and he walked slowly as though it ached to move.
That is the first paragraph of the story, and readers have no idea who "he" or "we" is, but readers can make a fairly educated guess as to the time of day. The big hint is "while we were still in bed." It could be making reference to an afternoon nap, but most people don't take those. It's likely early morning. Whoever is the "we" has slept most of the night and hasn't gotten up yet to start their day. The very early hours of the morning are often the coldest of the night too. This is why "he" would come in shivering and close the windows. He's trying to stay warm.
As for who the "we" is, readers who are parents are likely able to guess that the "we" is a mom and dad. Anecdotally, my own kids have quite regularly woken my wife or me up because of a "tummy ache" or because they are thirsty. Hemingway's opening paragraph to this story could sound mysterious and scary, but after having kids, I recognize the setup as something very normal for a parent.