Are photons ejected from atomic nuclei?
I have heard that electrons absorb or eject photons when transitioning from one orbital to another. Is this correct?


Not exactly.  The atom as a whole emits or absorbs the photon.  There is no reason to single out the electron versus the nucleus in such transitions. 


  Can atomic nuclei eject photons?


Yes there are two ways a nucleus in particular (as opposed to an atom taken as a whole) can emit photons.  

First, nuclei can exist in excited states. When an excited nucleus transitions to the ground state, a photon in the gamma or X-ray range of the electromagnetic spectrum is emitted.  Electrons do not have an analogous mechanism for emitting photons, because electrons are fundamental particles, while nuclei are not.   

Second, if a nucleus has a nuclear spin (S), then in a magnetic field there will exist  2S + 1 quantized energy states of the nuclei.  Transitions of nuclei between these states emit or absorb photons, the energy of which depends upon the strength of the field, for example in the radio frequency range.  This is the basis of NMR and MRI.  The electron has an analogous mechanism for emitting photons, known as ESR or EPR.