A new quantum-deformed model of anharmonic vibrations of diatomic molecules is proposed leading to the phenomenological Dunham expansion. The q-deformation of a general Hamiltonian is constructed from generators of the quantum group deformation of the algebra su(2). In different approximations it leads to Morse or Dunham results. The model parameters are obtained in terms of the experimental molecular constants which simplifies the applications. A novel physical interpretation for the deformation parameter q is obtained related to the third-order coefficient in the Dunham expansion. This work is supported in part by research grants from the Bulgarian NCSR F-1205/02 and CONACyT, Mexico.