They didn't single out Quirrell.

The wards of Hogwarts record that no foreign creature has entered, and that it was the Defense Professor who killed Hermione Granger.

"We're meant to conclude that the enemy has control of what the wards tell us. But that's actually something the enemy can only do with difficulty, or under special conditions; they're trying to create a false appearance of omnipotence."

.....

The old witch's eyebrows rose. "How did he identify you to the Hogwarts wards, then?"

A slight smile. "The Headmaster drew a circle, and told Hogwarts that he who stood within was the Defense Professor.

At first I thought he had a house-elf there, or his original body in some form (though snakes have no legs). Now I figure at least one of the N Defense Professors was a shrunken troll. (I just checked, and Lethifolds don't have legs either.)