One external conflict is that Julian's mother (and though he doesn't want to admit it, Julian too) has to cope with a new world in which black people have the same rights as white people. For example, now they can sit in the front of the buses with white people....
Julian's mother's racism and ingrained sense of her own superiority (which Julian shares) conflicts with a world that has begun to move beyond that mindset.
A second external conflict is between Julian and his mother. He hates her for clinging to the old ways of Southern graciousness and racism as if the world has not changed, and she, in turn, is angry at him for not taking the pride she would wish in his ancestry. He fears her open racism and how it might be displayed and wants her to change her outdated ways. This is generational conflict:
“They don't give a damn for your graciousness,” Julian said savagely. “Knowing who you are is good for one generation only. You haven't the foggiest idea where you stand now or who you are.”
She stopped and allowed her eyes to flash at him. “I most certainly do know who I am,” she said, “and if you don't know who you are, I'm ashamed of you.”
The story is told from Julian's point of view, so it is his internal conflicts that O'Connor explores. He is, as we find out at the end, conflicted between a deep love for his mother (he calls her "darling" as she is having her stroke) and hatred of all she represents. This hatred is really self hatred: he is a mirror of her though he doesn't want to admit it. For example, if she became a "martyr" sacrificing for him as a child, he now perceives himself as a martyr to her. He is St. Sebastian, ready to be pierced with arrows for her. As he takes her to her class,
He walked along, saturated in depression, as if in the midst of his martyrdom he had lost his faith.
While he tries to believe he is not like her, he shares her longing for their grandfather's gracious home, now lost:
He never spoke of it without contempt or thought of it without longing.
He also shares her snobbery: he wants to talk to black people, but only black people that he thinks are worthy of his time, such as doctors and lawyers.
Internally, Julian's conflict is that he must cope with his longing to be different from his mother and shake off her dominance while at the same time being like her in more ways than he wants to accept.