Let $\{X(t)= (X_1(t),X_2(t))^T,\ t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued continuous locally stationary Gaussian random field with $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_1, A_2 \subset \mathbb{R}^N$, precise asymptotic behavior of the excursion probability \[ \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty \] is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $ρ$.