Summary: We present a quasi-one-dimensional (Q1D) Bose-Fermi mixture model of strongly interacting Fermi gases through the confinement-induced resonance (CIR) to describe 3D fermions trapped in a Q1D geometry along the \(z\) axis. Based on the 1D Bethe ansatz method, we analyse the ground and excited states of the Q1D system and derive their integral equations in terms of the momentum distributions in the thermodynamic limit. By applying a Legendre expansion method, the crossover from the BCS superfluid of singlet pairs to the BEC of molecules is calculated in the ground and excited states. We then obtain the dependence of several physical quantities on the inverse of a dimensionless coupling parameter \(a_\bot /a\) with a dimensionless density parameter \(na_\bot\) of Q1D where \(a_{\perp}=\sqrt{\hbar/m\omega_{\perp}}\) is the transverse size and \(\omega_\bot\) is the 2D harmonic trap frequency in the xy plane. Our results can go back to the smooth 1D crossover when \(na_\bot \rightarrow 0\). The theoretical results quantitatively show that the strongly coupling Fermi gas in the Q1D behaves as a Bose-Fermi mixture near the crossover.