The interaction of molecules with light may lead to electronic transitions and simultaneous vibrational excitations. Franck-Condon factors (FCFs) play an important role in quantifying the intensities of such vibronic transitions occurring during molecular photoexcitations. In this article, we describe a general method for estimating FCFs using a quantum information processor. The method involves the application of a translation operator followed by the measurement of certain projections. We also illustrate the method by experimentally estimating FCFs with the help of a three-qubit nuclear magnetic resonance quantum information processor. We describe two methods for the measurement of projections: (i) using the Moussa protocol and (ii) using diagonal tomography. The experimental results agree fairly well with the theory.