Quirrel knows the killing curse better than Harry does. He also is likely to have information about Harry, which Harry lacks.
So the fact that he is sure Harry could pull off the killing curse does not mean that Harry has any reason whatsoever to share his confidence. And having it fizzle would be fatal.
“Finite the rock” on the other hand is going to work. So he went with that.
Quirrel knows the killing curse better than Harry does. He also is likely to have information about Harry, which Harry lacks. So the fact that he is sure Harry could pull off the killing curse does not mean that Harry has any reason whatsoever to share his confidence. And having it fizzle would be fatal. “Finite the rock” on the other hand is going to work. So he went with that.