That is a tough question actually.
If I am not completely mistaken gravitational waves do not cause gravitational time dilation in principle (time components in g are unaffected). However they do change the spatial geometry.
But I think the wave would have to have a frequency comparable to atomic transition energies to affect the quantum chemistry of atoms in the atomic clock. I'm not sure.
If you measure time by dividing distance over speed of light then it should affect your measurement if it is quick enough because distances do change