Quirrell and Harry are both horcruxes of Voldemort, and there is a decent chance that Quirrell has guessed that this is the case by now, if he didn't always know. Quirrell thus has a very good reason to be nice to Harry...they are partially the same person.

But just how much similarity does hpmor Voldemort bear to cannon Voldemort?

Intelligence boost aside, both Harry and Quirrell have the exact same motives as canon Voldemort (power and immortality). The only difference between them is that Harry has an ethical component to his utility function - that's pretty much the only difference between Harry and Quirrell. Tom Riddle for his part is not against ethics - he just doesn't care about them. There are different varieties of evil: let's not confuse amorality with sadism.

So there is absolutely no reason why Quirrell should view Harry as an enemy, except where Harry interferes with his plans because of his morality. If Harry succeeds at all his goals, so does Quirrell (to some extent. There is still the "dominance" component of power, which is a zero sum game. It's hard to tell how much Quirrell cares about that.)

Harry's view of Quirrell is slightly more problematic. Because of Quirrell's lack of ethics constraints, Quirrell has many more options open to achieve his power/immortality goal than Harry does. So while Harry doesn't need to kill Quirrell, he does need to prevent him from achieving is goals in unethical ways.

In fact, my current prediction is that Harry will "win" by achieving Quirrell's goals ethically, thereby making it unnecessary for Quirrell to behave immorally.