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.