In transition metal oxides, magnetic dipole-dipole (DD) and chiral Dzyaloshinsky-Moriya (DM) interactions between nearest neighboring spins are comparable in magnitude. In particular, the effects of the DD interaction on the physical properties of magnetic nanosystems cannot be simply neglected due to its long-range character. For these reasons, we employed here a new quantum simulation approach in order to investigate the interplay of these two interactions and study their combined effects upon the magnetic vortical structures of monolayer nanodisks. Consequently, we found out from our computational results that, in the presence of Heisenberg exchange interaction, a sufficiently strong DD interaction is also able to induce a single magnetic vortex on a small nanodisk; a strong DM interaction usually gives rise to a multi-domain structure which evolves with changing temperature; In this circumstance, if a weak DD interaction is further considered, the multi-domains merge to form a single vortex in the whole magnetic phase. Moreover, if only the Heisenberg exchange and chiral DM interactions are considered in simulations, our results from calculations with different spin values show that the transition temperature TM is simply proportional to S(S+1); if the temperature is scaled with TM, and the calculated magnetizations are divided by the spin value S, their curves exhibit very similar features in the whole temperature region below TM.