We consider a gravitational perturbation of the Jackiw-Teitelboim (JT) gravity with an arbitrary dilaton potential and study the condition under which the quadratic action can be seen as a -deformation of the matter action. As a special case, the flat-space JT gravity discussed by Dubovsky et al. [arXiv:1706.06604] is included. Another interesting example is a hyperbolic dilaton potential. This case is equivalent to a classical Liouville gravity with a negative cosmological constant and then a -deformation of the matter action with a finite coupling is realized as a gravitational perturbation on AdS2.