We define generalized function spaces $LG'_0(\mathbf{R}^2_+)$ and $LG'_e(\mathbf{R}^2_+)$ over the quarter-plane and study the convolution type equations by using the Laguerre expansions of their elements in two dimensions. The applications on partial integro-differential equations of the form \begin{align*} f(t,s)=cǎrphi(t,s)&+ıt^ıfty_0k_1(t-au)ǎrphi(au,s)dau+ıt^ıfty_0k_2(s-igma)ǎrphi(t,igma)digma &+ıt^ıfty_0ıt^ıfty_0k(t-au,s-igma)ǎrphi(au,igma)dau\,digma \end{align*} are given.