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The transverse momentum distributions of final-state particles are very important for high energy collision physics. In this work, we investigate J/ψ and Υ meson distributions in the framework of a particle-production source, where Tsallis statistics are consistently incorporated. The results are in good agreement with the experimental data in p-p and p-Pb collisions at LHC energies. The temperature of the emission source and the nonequilibrium degree of the collision system are extracted.