The aim of this work is the derivation of two approximated expressions for the two dimensional Gaussian Q-function, $Q(x,y;ρ)$. These expressions are highly accurate and are expressed in closed-form. Furthermore, their algebraic representation is relatively simple and therefore, convenient to handle both analytically and numerically. This feature is particularly useful for two reasons: firstly because it renders the derived expressions useful mathematical tools that can be utilized in numerous analytic performance evaluation studies in digital communications under fading; secondly because the two dimensional Gaussian Q-function is neither tabulated nor a built-in function in popular mathematical software packages such as $Maple$, $Mathematica$ and $Matlab$.