I believe Snape related to Dumbledore what he observed about Harry exhibiting pain from his scar during the opening day feast (pg. 126). Upon hearing this, Dumbledore told him to keep an eye on Quirrell.

Yeah, Dumbledore had to have known something was going on with Quirrell which is why he asked Snape to spy on him. Although, it is hard to believe that Dumbledore would make the path to get to the mirror so easy that three first years would be able to get to it!!

Snape did not report to Dumbledore because he suspected him for most of the book and only really knew towards the end. Even when he did know who would have believed him? Snape used to be a death eater and Quirrel looked like an innocent frightened teacher.

I just think that he wanted to get quirrell 4 himself.

Dumbledore suspected him.. Snape knew that,as he told him. But they didn't really know,they only suspected him.

Maybe he wanted to tackle the situation by himself.

doesn't he have to appear loyal to voldemort as part of his double identity?

Dumbledore did know! He says in the Pensieve, after Snape dies, he tells Snape to keep an eye on Quirrell!

This may have helped the case about Quirrel but I can't remember if Hagrid ever spoke up about seeing Quirrel the day they went to Gringotts and the Stone was nearly stolen? Regardless, I think that Snape had told, that's just who he is, and Dumbledore tried to keep him from doing anything rash.

I'd always wondered why Snape never told anyone but never really thought too much about it because I was young when I first read the book and we always just saw things from Harry's point of view.

I hadn't thought of that.

Even with them suspecting Quirrell, because he kept up his act so well, perhaps Dumbledore and Snape felt they might be able to diffuse the situation and convince him to stop what he was doing? I think that's what I would do if I knew something was up but had no concrete evidence. Because what others have been saying is right, without concrete evidence they couldn't really do anything and it would only lead to making them look paranoid.

I don't think Snape suspected Quirrel, in the Seventh book(Deathly Hallows) it was revealed that Dumbledore ordered Snape to keep an eye on Quirrel.

He had no proof... it was just his "instinct", however he was right!