The Bohr's model of the atom was defined only for hydrogen and hydrogen-like species. Yet, the Franck-Hertz experiment was performed with Mercury vapour- which, obviously, isn't hydrogen-like. What's more intriguing is that the observations made were in good agreement with Bohr's model. Take a look here: http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html Doesn't this mean that Bohr's model applies to other atoms too? But it is known that one of the major limitations of the model was that the structure of non hydrogen-like atoms couldn't be explained. Help, folks?
The Franck-Hertz experiment and the Bohr model had very different aims so they do not conflict.
The Franck-Hertz experiment just shows that excited states exist i.e. it shows that there are certain electron energies where the scattering of electrons is strong because the electron energy corresponds to a spacing between two energy levels. This is true for all atoms since all atoms have energy levels with excitations between them. Mercury just happened to be convenient for doing the experiment. However the Franck-Hertz experiment just observes the energy spacing and makes no attempt to explain or predict it.
The Bohr model is more ambitious because it attempts to explain not only why energy levels exist, but also the spacing between them. The problem is that while this works well for atoms with a single electron it fails for multi electron atoms. That doesn't mean it predicts excited states don't exist for atoms like mercury, only that the Bohr model cannot explain them. That is it cannot predict the energies of the peaks in the FH graph.