I think Quirrell wants to teach Harry to give up.

First he does it explicitly, during the lesson. Okay, that is not really giving up; it's rather "pretending to lose, so you survive and have your revenge later". But even that is difficult for Harry. So maybe learning this lesson is just a first approximation towards really giving up?

"The boy cannot be allowed to continue in this state of mind. He will become dangerous. ... It is my professional judgment ... that the boy could join the ranks of those whose rituals are inscribed upon the tombstones of countries."

And recently it seems like Quirrell is taking his time to break Harry psychologically, demonstrating his advantage by overpowered attacks, reminding him about all the hostages he can kill.

The sadness and horror that Harry had pushed down flared up again, and his dark side had no stored patterns for handling the emotions. Why, Professor Quirrell, why are you like this...

Professor Quirrell smiled. "That reminds me. Have you betrayed me yet?"

It seems as if Quirrell wants Harry to betray him. And then he would kill the hostages. And then he would tell Harry: "It's all your fault."

Which would be the opposite of what Hermione told Harry. But we already know that Quirrell is trying to push Harry in the opposite direction than Hermione would.

What is he trying to achieve by this? Maybe if Harry gives up, he will lose his ability to cast Patronus 2.0. Why is that important? Because it is the only spell that Harry knows and Quirrell does not; therefore the greatest danger to Quirrell?

Or maybe Quirrell wants Harry to snap and become an efficient ruthless killing machine just like him.