We propose a renormalizable $B-L$ Standard Model (SM) extension based on $Q_6$ symmetry which accommodates fermion mass and mixing parameters. Both normal and inverted neutrino mass orderings, the smallness of the active neutrino masses are generated through type I seesaw mechanism. The obtained physical parameters are well consistent with the global fit of neutrino oscillation [1] while the quark masses are in good agreement with the recent experimental data [2]. The model also predicts effective neutrino mass parameters of $\langle m_{ee}\rangle =3.23\times {10}^{-3}\text{eV}$, $m_{\beta }=8.92\times {10}^{-3}\text{eV}$ for normal hierarchy (NH) and $\langle m_{ee}\rangle =m_{\beta }=4.96\times {10}^{-2}\text{eV}$ for inverted hierarchy (IH) which are all well consistent with the recent experiments.