Learning parameters of noise and filter coefficients from data where data and noise both have Gaussian distributions
Your problem is that of multivariate linear regression. Finding each row of $A$ corresponds to a classical linear regression problem, which can be solved using the method of least squares, which is optimal here since the noise is Gaussian.

Once you have an estimate for $A$, you can estimate the parameters of $N$ by isolating $N = Y-AX$. (This is not necessarily the optimal method since you only have an estimate for $A$, but practically it might be good enough.)