  Colour $SU(3)$ group is an exact symmetry of Quantum Chromodynamics, which
describes strong interactions between quarks and gluons. Supplemented by two
internal symmetries, $SU(2)$ and $U(1)$, it serves as the internal symmetry of
the Standard Model, describing as well the electroweak interactions of quarks
and leptons. The colour$SU(3)$ symmetry is exact, while two other symmetries
are broken by means of the Higgs-Kibble mechanism. The three colours and
fractional quarks charges with values $1/3$ and $2/3$ suggest that the cyclic
group $Z_3$ may play a crucial role in quark field dynamics. In this paper we
consequently apply the $Z_3$ symmetry to field multiplets describing colour
quark fields. Generalized Dirac equation for coloured $12$-component spinors is
introduced and its properties are discussed. Imposing $Z_3$-graded Lorentz and
Poincar\'e covariance leads to enlargement of quark fields multiplets and
incorporates additional $Z_2 \times Z_3$ symmetry which leads to the appearance
of three generations (families) of distinct quark doublets.
