We're told so during their conversation in the woods.

No, there is only one person who holds so much power over you, and who would be most perturbed to find you executing any plot without his knowledge. Your true and hidden master, Albus Dumbledore."

"What?" hissed the Potions Master, the anger plain upon his face.

"But now, it seems, you are moving on your own; and so I find myself most intrigued as to what you could possibly be doing, and why."

Quirrell may not know about his love for Lily, but I consider that highly unlikely, since Snape apparently still asked Voldemort not to kill her. Quirrellmort would certainly have put two and two together by now.

But that passage explicitly tells us that Quirrell knows that Snape is no longer acting under Dumbledore's orders.