We study the elastic energy landscape of two-dimensional tin oxide (SnO) monolayers and demonstrate a transition temperature of T_{c}=8.5±1.8 K using ab initio molecular dynamics (MD) that is close to the value of the elastic energy barrier J derived from T=0 K density functional theory calculations. The power spectra of the velocity autocorrelation throughout the MD evolution permit identifying soft phonon modes likely responsible for the structural transformation. The mean atomic displacements obtained from a Bose-Einstein occupation of the phonon modes suggest the existence of a quantum paraelastic phase that could be tuned with charge doping: SnO monolayers could be 2D quantum paraelastic materials with a charge-tunable quantum phase transition.