in the 2D case, you can write $X=AZ$, where $A$ is a $2\times2$ nonsingular real matrix, $Z:=[Z_1,Z_2]^T$, and the $Z_j$'s are iid standard normal. You want to find $$p:=P(n_1\cdot X>0,\;n_2\cdot X>0),$$ where $n_1$ and $n_2$ are unit vectors in $\mathbb R^2$ and $\cdot$ is the dot product.

By the rotational symmetry of the distribution of $Z$, you have $$p=P(m_1\cdot Z>0,\;m_2\cdot Z>0)=\frac1{2\pi}\arccos\frac{m_1\cdot m_2}{|m_1|\,|m_2|},$$ where $m_j:=A^T n_j$ and $|\cdot|$ is the Euclidean norm on $\mathbb R^2$.

In the 2D case, you can write $X=AZ$, where $A$ is a $2\times2$ nonsingular real matrix, $Z:=[Z_1,Z_2]^T$, and the $Z_j$'s are iid standard normal. You want to find $$p:=P(n_1\cdot X>0,\;n_2\cdot X>0),$$ where $n_1$ and $n_2$ are unit vectors in $\mathbb R^2$ and $\cdot$ is the dot product.

By the rotational symmetry of the distribution of $Z$, you have $$p=P(m_1\cdot Z>0,\;m_2\cdot Z>0)=\frac1{2\pi}\arccos\frac{m_1\cdot m_2}{|m_1|\,|m_2|},$$ where $m_j:=A^T n_j$ and $|\cdot|$ is the Euclidean norm on $\mathbb R^2$.

When the dimension is $>2$, the problem similarly reduces to finding the probability that a standard normal random vector is in a polyhedral cone. This is a difficult problem, admitting a certain recursive solution, which can be resolved more or less explicitly for dimensions $\le4$. See e.g. Plackett and references there, notably to Schläfli.