  The systematic study of isoscalar (IS) and isovector (IV) giant quadrupole
responses (GQR) in normal and superfluid nuclei presented in [G. Scamps and D.
Lacroix, Phys. Rev. 88, 044310 (2013)] is extended to the case of axially
deformed and triaxial nuclei. The static and dynamical energy density
functional based on Skyrme effective interaction are used to study static
properties and dynamical response functions over the whole nuclear chart. Among
the 749 nuclei that are considered, 301 and 65 are respectively found to be
prolate and oblate while 54 do not present any symmetry axis. For these nuclei,
the IS- and IV-GQR response functions are systematically obtained. In these
nuclei, different aspects related to the interplay between deformation and
collective motion are studied. We show that some aspects like the fragmentation
of the response induced by deformation effects in axially symmetric and
triaxial nuclei can be rather well understood using simple arguments. Besides
this simplicity, more complex effects show up like the appearance of
non-trivial deformation effects on the collective motion damping or the
influence of hexadecapole or higher-orders effects. A specific study is made on
the triaxial nuclei where the absence of symmetry axis adds further complexity
to the nuclear response. The relative importance of geometric deformation
effects and coupling to other vibrational modes are discussed.
