  We give a parametric representation of the effective noncommutative field
theory derived from a $\kappa$-deformation of the Ponzano-Regge model and
define a generalized Kirchhoff polynomial with $\kappa$-correction terms,
obtained in a $\kappa$-linear approximation. We then consider the corresponding
graph hypersurfaces and the question of how the presence of the correction term
affects their motivic nature. We look in particular at the tetrahedron graph,
which is the basic case of relevance to quantum gravity. With the help of
computer calculations, we verify that the number of points over finite fields
of the corresponding hypersurface does not fit polynomials with integer
coefficients, hence the hypersurface of the tetrahedron is not polynomially
countable. This shows that the correction term can change significantly the
motivic properties of the hypersurfaces, with respect to the classical case.
