We investigate the energy-momentum tensor of pseudoscalar charmonium and bottomonium within the framework of the self-consistent light-front quark model. The gravitational form factors (GFFs), namely the $A$ and $D$-terms, are evaluated in terms of the light-front wave functions. The corresponding spatial mechanical distributions in the transverse plane are obtained through the Fourier transform of these GFFs. To examine the sensitivity of the results to the internal quark-antiquark distribution inside the meson, two distinct Gaussian forms are employed for the spatial part of the wave function. We analyze several mechanical properties in the transverse plane, including the momentum density, pressure distribution, shear stress, force density, and internal energy density. The pressure distribution exhibits a node where it changes sign from positive (repulsive) to negative (attractive) with increasing transverse distance. The force distribution remains positive throughout the transverse plane, supporting the stability condition proposed in earlier studies. Most of the spatial distributions, except for the shear stress, are found to be sensitive to the choice of the spatial wave function near the center of the meson, while they become nearly insensitive toward the periphery. In contrast, the shear stress distribution exhibits noticeable sensitivity to the choice of wave function in the intermediate transverse region.