With the increase of integration scale, heat dissipation becomes one of the major problems in high density electronic devices and circuits. Controlling and reusing the heat energy in such miniaturized structures are essential topics for current and future technologies. With the development of microfabrication technology and low-temperature measurement technology in the last two decades, the thermoelectric measurement in low-dimensional sample has been feasible, and the thermal transport has received more and more attention. For the multi-terminal device, there is a novel thermoelectric phenomenon, called the spin Nernst effect, in which spin currents (or spin voltages) are generated perpendicularly to the temperature gradient. The spin Nernst effect has been confirmed experimentally, and has been theoretically studied in a variety of materials. In this paper, the spin and charge Nernst effect in a pair of vertically aligned quantum dots attached to four leads are studied in the Coulomb blockade regime based on the nonequilibrium Green's function technique. We focus on the influences of magnetic configuration and intra-dot (inter-dot) Coulomb interaction on the spin and charge Nernst effect. It is found that the signs and the magnitudes of spin and charge Nernst effect can be modulated by adjusting the magnetization directions of ferromagnetic electrodes. When the magnetic moments in the 1 and 3 electrodes are turned to antiparallel alignment, the pure spin Nernst (without charge Nernst) effect can occur by applying a transverse temperature gradient. Conversely, the spin and charge Nernst effect disappear if the magnetic moments of lead 1 and lead 3 are in the case of parallel configuration. Except for left and right thermal leads, we investigate the effect of the middle lead (lead 4) on the property of the Nernst effect. We find that when the normal metal lead 4 is transferred to ferromagnetic metal, the spin and charge Nernst effect both can be obtained simultaneously. In the end of the paper, we study the influences of intra-dot and inter-dot Coulomb interaction on the spin dependent Nernst coefficient. Through numerical calculations, we demonstrate that the magnitude of the Nernst effect is less dependent on the polarization strength of ferromagnetic electrodes, but can be remarkably enhanced by the Coulomb blockade. The spin Nernst coefficient is predicted to be more than two orders of magnitude larger than that of the case of zero Coulomb interaction. All the results indicate that the proposed four-terminal double quantum dot nano system is a promising candidate for spin caloritronic device.