The conflict could be Snape's pride. He is a proud man and asking for help, especially from a student like Hermione, goes against his nature. So he constantly battles his own ego while Hermione has to deal with his snarky remarks.
There could be a conflict where Hermione is hesitant to help Snape because of his past treatment of her and her friends. She has to decide whether to put aside her personal feelings for the greater good, like helping him with a brewing threat to Hogwarts. And Snape has to convince her that he truly needs her help.
Another possible conflict is that Snape's need for Hermione's help is related to something that is against Hogwarts rules. Hermione, being a rule - follower, has to decide if she will break the rules to assist him. Snape, on the other hand, is desperate for her help and tries to persuade her that the end justifies the means.
Maybe Snape has a difficult spell he can't seem to master, and as much as he hates to admit it, Hermione's academic prowess makes her the best person to ask for help. He approaches her in a very snarky way, but she sees through his facade and decides to assist him. This could lead to an interesting dynamic where they start to respect each other more.
One possible reason could be related to a difficult potion she's trying to brew. Snape is a potions master, and Hermione might be facing an ingredient or a process that she can't figure out on her own. For example, a rare potion that requires precise handling of dangerous ingredients.
Maybe Snape is involved in some sort of mystery or danger within Hogwarts that has to do with magic spells. Hermione, with her vast knowledge of spells and her in - depth study of the Hogwarts library, could be a great help to him in deciphering old spells or magical texts related to the situation.
Perhaps Hermione is injured in a fight against dark wizards while on a secret mission. Snape discovers her and, despite their past differences, decides to help. He uses his skills in dark magic knowledge to counteract the dark spells that have injured her. This could lead to a begrudging respect between them as they work together to heal her and keep the situation a secret from other Hogwarts students and staff.
Maybe she would say something like 'Snape, your constant glares don't intimidate me. I've faced far worse than you in the library stacks.' Here, Hermione is using her love for the library and her knowledge - seeking adventures as a way to snarkily put Snape in his place, implying that his attempts at intimidation are nothing compared to what she has endured in her academic pursuits.
She might be required to research in the restricted section of the library. Given her penchant for knowledge and Snape's often - dark and esoteric interests, the restricted section could hold the key to whatever he needs help with.
He could also be a bit sarcastic in his response. Snape has a sharp tongue, so he might say something like 'Miss Granger, what makes you think I have the time or inclination to help you?' But as Hermione explains her situation, he might gradually soften, realizing the seriousness of the matter. Maybe he starts to see a bit of himself in Hermione's thirst for knowledge.
Snape could be very sarcastic and cold at first. He might taunt Harry about finally coming to him for help. However, as he realizes the gravity of the situation, he could start to take it more seriously and actually use his skills and knowledge to help Harry. For example, he might say something like 'Potter, so you've finally realized that you're in over your head and need my help? Well, don't expect me to hold your hand.' But then he would still start to investigate the problem.
Snape would probably be very cold and indifferent at first. He might even think it's some sort of Gryffindor ploy to get his attention or make him look bad. He has always had a low opinion of Hermione's house, so his initial reaction would be guarded.
Maybe Harry discovers something about his past, perhaps related to his parents, that only Snape can help with. Snape was closely associated with Harry's parents. He might know some secrets or have some information that could help Harry understand a family mystery or some hidden power within him.