The transverse momentum distributions of final-state particles are very important for high energy collision physics. In this work, we investigate $J/\psi$ and ${\rm Y}$ 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 $p\text{-}p$ and $p$-Pb collisions at LHC energies. The temperature of the emission source and the nonequilibrium degree of the collision system are extracted.