http://www.environment.alberta.ca/apps/basins/
Haven't looked myself, but general rule of thumb is that while you can certainly fish the Athabasca during June-July-August, it gets daily flows of glacial water and glacial silt / flour. That's aside from any rain or snowmelt.
Bottom line is that the river doesn't clear up until September, that's usually a better month for clarity. October and April even better than that. Nov-March is clear, but the weather is obviously colder.
Good luck.
Smitty