Q-factor of external bearing BBs
Not necessarily. 

For one thing, many cartridge style bottom brackets will have some spindle exposed when the crank is mounted. This is because even though two bottom brackets might both be 68 mm, one may have a longer spindle than the other. If you mount the same crank on both bottom brackets, it might fit flush against the bottom bracket shell on the shorter spindle, but would have a gap on the longer spindle. 

More importantly, most cranks curve outward from the bottom bracket so that the cranks will clear the chainstays. You can see what I mean in this picture:



That's obviously an external bottom bracket, but a crank that sat flush against the bottom bracket shell would have to have an even more significant curve to clear the chain stays (assuming the crank arms were the same length, thickness, etc).

Technically, Q-factor refers specifically to the distance between the point where the pedals attach to the crank arms.



However, many people refer to Q-factor in the broader sense of the distance between your feet when pedaling. Given this broader definition, your Q-factor would also be affected by the length of your pedal spindles combined with the position of your feet on the pedals, whether you're clipped in or riding on platforms.