Molecules trapped inside fullerenes exhibit interesting quantum behavior, including quantization of their translational degrees of freedom. In this study, a theoretical framework for predicting quantum properties of nonlinear small molecules in nonsymmetric open-cage fullerenes (OCFs) has been described along the lines of similar theories which treat small molecules inside C-60 and clathrate cages. As an example, the coupled translational-rotational energy structure has been calculated for the case of CH4 inside a known OCF. The calculated energy levels have been used to calculate the equilibrium fraction of incorporated CH4 as well as the translational heat capacity for the encapsulated molecule. The heat capacity shows an anomalous maximum at 239 K for CH4 and 215 K for CD4 which are not present in free methane.