Quadrant probability of non-centric bivariate normal distribution

Question

Suppose $(X,Y)$ has a bivariate normal distribuion with non-zero mean vector $\mu$ and covariance matrix $\Sigma$. What should $\mathbb{P}(X>0,Y>0)$ be? My attempt gives me an definite integral $\int^\infty_c \phi(y) \Phi(\frac{y-a}{b}) dy$ for some constants $a,b,c$ at some point, where I am unable to continue. Therefore any suggestion to solving the probability or the way in solving the integral will be appreciated.