The intramolecular radiative and nonradiative relaxation processes of three thiophene-S,S-dioxide derivatives with different molecular rigidity are investigated in different solutions and in inert matrix. We show that the fluorescence quantum efficiency and the relaxation dynamics are strongly dependent on the environment viscosity, whereas they are almost independent of the environment polarity. We demonstrate that this strong dependence is due to an environment dependent nonradiative decay rate, whereas no relevant variations of the radiative decay rate are observed. We demonstrate that the dipole coupling with the solvent does not provide an efficient nonradiative decay channel and that the S(n) - S(1) vibrational relaxation is very efficient in all of the molecules and for all of the investigated environments. Moreover first-principles time-dependent density-functional theory calculations in the correct, i.e., excited-state, molecular conformation, suggest that significant contributions of intersystem crossing to the triplet manifold can be excluded. We then conclude that the main nonradiative process determining the fluorescence quantum efficiency of this class of molecules is S(1) - S(0) internal conversion (IC). An explanation for the IC rate dependence in terms of the environment viscosity, molecular rigidity, S(1) - S(0) energy-gap, and molecular volume is presented.