In a particle-γ coincidence experiment, a thick tungsten target, of natural isotopic abundance, was bombarded with a and 16O beams. From analysis of the deexcitation γ-rays following Coulomb excitation, the spectroscopic quadrupole moment of the second 2 + state (the 2†' state) was determined for 186W and 184W. In a separate Coulomb excitation experiment a thin, isotopically enriched 186W target was bombarded with 16O ions. From analysis of projectiles scattered elastically and inelastically the quadrupole moment of the 2 +' state of 186W was extracted. The results of the two experiments are in good agreement. The quadrupole moment of the 2 +' state is found to be opposite in sign to that of the first 2 + state for both isotopes studied. However, its magnitude decreases rapidly in going from 186W to 186W, in contrast to the predictions of the rotation-vibration or asymmetric rotor models. The microscopic theory of Kumar and Baranger does predict the experimental trend, qualitatively. Thus the present results are interpreted as being evidence of strong coupling between β and γ degrees of freedom in the tungsten isotopes, which, according to the theory of Kumar and Baranger, is the source of the reduced value of the quadrupole moment.