Monthly Updates can include new driver files inside of them, but it's rarely done.
MS prefers to push individual driver updates using the normal driver check process. They will make an exception and try to force critical updates for a few drivers that are very common (like for Intel or AMD), so everyone has the same minimum version for stability.
It doesn't happen often and won't include any more complicated drivers (like for a NVIDIA or AMD GPU). I hear a lot of people say the Quality Updates setting is important, but I doubt it. The first two are the ones that really matter. Even if the Monthly CU includes a driver, it will be ignored if you have a newer driver than what's included in the KB.