The paper puts forward an extension of the Pythia8 software package, which makes it possible to calculate collisions of longitudinally polarized protons with the production of J/ψ mesons. The transverse momentum spectra of J/ψ mesons and the double longitudinal spin asymmetry of them at an energy of 510 GeV have been obtained. The calculated results were compared with experimental data, and a satisfactory agreement between them was established. This success served as the basis for predicting the results of the SPD experiment at the NICA collider. The corresponding calculations of the transverse momentum spectrum and the double longitudinal spin asymmetry of the J/ψ meson at an energy of 27 GeV were presented. All conclusions were based on the collinear parton model.