I'm solving incomprssible Navier-Stokes equations with artificial compressibilty method. In order to use large pesudo time step, I use implicit scheme to integral in time. The converged solution, where all the dq terms drop to machine zero, is found to be dependent on cfl number. Is this normal? If there are any problems, what could the problems be?

Solutions of some schemes such as Macormack, Strang type splitting, really dependent on time step. You will be required to give specific scheme used so that others can answer you.

The answer is: when you have reduced the time step to the point where your solution no longer changes much, you will have properly modeled your system.

It is vitally important to make sure that your solutions are properly converged in time and in space.

Implicit schemes guarantee stability, not accuracy, in time.

what is the problem you are working for?

Maybe the problem is sensitive to disturbence.