  In this article, we get the spectral solution $\mathcal{P}_{\mu}^{m}$ of
operators $m(\mathcal{L}, -\Delta_\mathfrak{z})$, the joint functional calculus
of the sub-Laplacian and Laplacian on the centre of M\'etivier group. Then, we
give some group-analogues of the Thomas-Stein-type restriction theorem,
asserting the mix-norm boundness of the restriction operators
$\mathcal{P}_{\mu}^{m}$ for two classes of functions
$m=(a^\alpha+b^\beta)^\gamma$ and $m=(1+a^\alpha+b^\beta)^\gamma$ with $\alpha,
\beta>0, \gamma\neq0$.
