They might start by arguing as they have different personalities and beliefs from their time at Hogwarts. But then, they could gradually discover new sides of each other. Maybe Draco would share his insecurities about his family's reputation, and Hermione would listen with empathy.
It's possible that they initially have an awkward encounter during their alone time. However, as time passes, they might find common interests like a love for certain magical creatures or a shared concern for the state of Hogwarts. They could then form an unexpected friendship.
During their alone time, they could end up working together on a project, perhaps something related to the wizarding world's future. Hermione's intelligence and Draco's knowledge of the pure - blood society could combine in interesting ways.