Driven quantum systems subject to non-Markovian noise are typically difficult to model accurately. Recently, we presented a systematic method based on a generalized cumulant expansion for deriving pseudo-Lindblad Master equations (PLMEs) that can accurately capture the effects of classical, highly correlated noise. PLMEs are equations of motion that parallel the standard Lindblad form, however, allow for the jump rates to be negative. In this talk, we expand on our previous efforts, and show how to treat noise that is not just highly correlated, but also quantum. Using techniques based on Floquet theory, we also discuss how to capture the effects of strong coherent driving, that were previously not discussed. Through numerical simulations, we show that our effective description of the system evolution can often lead to a more accurate prediction of its dynamics than can be achieved by commonly utilized phenomenological approaches.