In this paper, we begin from the Klein-Gordon (KG) equation in 2D and change the standard partial derivatives by the Dunkl derivatives to obtain the Dunkl-Klein-Gordon (DKG) equation. We show that the generalization with Dunkl derivative of the z-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we show that DKG equations for the 2D Coulomb potential and the Klein-Gordon oscillator are exactly solvable. For each of the problems, we find the energy spectrum from an algebraic point of view by introducing suitable sets of operators which close the su(1, 1) algebra and use the unitary theory of representations. Also, we find analytically the energy spectrum and eigenfunctions of the DKG equations for both problems. Finally, we show that when the parameters of the Dunkl derivative vanish, our results are suitably reduced to those reported in the literature for these 2D problems.