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.
EDIT: New discussion thread here.
This is a new thread to discuss Eliezer Yudkowsky's Harry Potter and the Methods of Rationality and anything related to it. With two chapters recently the previous thread has very quickly reached 500 comments. The latest chapter as of 17th March 2012 is Ch. 79.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author's Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven, eight, nine, ten.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: