If $\varrho$ is not zero there isn't much you can do. If you have an implementation of the bi-variate normal CDF $\Phi_2(x,y,\varrho)$ you can "simplify" it to $$ \mathbb P\Big(X<\Phi^{-1}(p),Y<\Phi^{-1}(p)\Big)=\Phi_2\Big(\Phi^{-1}(p),\Phi^{-1}(p),\varrho\Big)\,. $$ This RHS is known as Gauss copula. If $p=1$ and $\Phi^{-1}(p)=+\infty$ the RHS is one.

Needless to say that for $\rho=0$ the RHS is $\Phi(\Phi^{-1}(p))\,\Phi(\Phi^{-1}(p))=p^2\,.$

If $\varrho$ is not zero there isn't much you can do. If you have an implementation of the bi-variate normal CDF $\Phi_2(x,y,\varrho)$ you can "simplify" it to $$ \mathbb P\Big(X<\Phi^{-1}(p),Y<\Phi^{-1}(p)\Big)=\Phi_2\Big(\Phi^{-1}(p),\Phi^{-1}(p),\varrho\Big)\,. $$ This RHS is known as Gauss copula. If $p=1$ and $\Phi^{-1}(p)=+\infty$ the RHS is one.

Needless to say that for $\varrho=0$ the RHS is $\Phi(\Phi^{-1}(p))\,\Phi(\Phi^{-1}(p))=p^2\,.$