This is also discussed and shown in "Computational Structural Engineering. Automatic calculation of mechanical structures" by C. Gianini. This book uses examples from Abaqus. One of the tests clearly shows the difference caused by mesh refinement:
(...) lower mesh density in a finite element model tends to overestimate the stiffness of the structure. (...) we can see how the model with coarse mesh tends to have, generally, higher frequency values than the corresponding model with fine mesh, a clear indication that, mass being equal, the calculated stiffness is higher in the first case.
Also in "Building Better Products with Finite Element Analysis" by V. Adams it's advised to pay attention to mesh density when performing modal analyses:
(...) as with a displacement analysis, an overly coarse mesh will result in an overly stiff structure and fictitiously high modes. A modal analysis must be converged as any other structural study.