He already tried earlier at Gringotts. For Hogwarts he first needed to find out how to overcome the other defenses.
Quirrell himself admits the attempted theft from Vault 713 in the final confrontation:
"When I failed to steal the stone from Gringotts, he was most displeased. He punished me ... decided he would have to keep a closer watch on me." -- Harry Potter and the Philosopher's Stone, Chapter 17 "The man with two faces"
As already mentioned above by Cadence, how to overcome the defenses at Hogwarts was something he first needed to find out:
"You're too nosy to live, Potter. Scurrying around the school on Halloween like that, for all I knew you'd seen me coming to look at what was guarding the Stone." -- Harry Potter and the Philosopher's Stone, Chapter 17 "The man with two faces"
Quirrell's questionable enquiries didn't go unnoticed. Albeit interpreted completely wrong by Harry, the little talk between Snape and Quirrell he overheard documents that at least Snape was aware of Quirrell's snooping:
Snape interrupted him: "Have you found out how to get around the beast of Hagrid's yet?" -- Harry Potter and the Philosopher's Stone, Chapter 13 "Nicholas Flamel"
As Harry and his friends become aware, Quirrell indeed does find out how to calm cute Fluffy:
"Well -- yeah -- how many three-headed dogs d'yeh meet, even around Hogwarts? So I told him, Fluffy's a piece o' cake if yeh know how to calm him down, jus' play him a bit o' music an' he'll go straight off ter sleep --" Hagrid suddenly looked horrified. "I shouldn'ta told yeh that!" he blurted out. -- Harry Potter and the Philosopher's Stone, Chapter 16 "Through the Trapdoor"
It is unclear how much Quirrell found out about the defenses and what was still unknown to him. At minimum he didn't know how to retrieve the stone from the Mirror of Erised. He likely would have preferred to find out still more, but Voldemort, the possibility of the stone being relocated again or even destroyed, and the nearing end of the school year forced him to act.