In this paper, the authors considered the following quasilinear chemotaxis model with nonlinear sensitivity, Lotka-Volterra competitive kinetics and nonlinear signal production \begin{align*} &u_t=\nabla\cdot(D_1(u)\nabla u)-\nabla\cdot(S_1(u)\nabla v)+\mu_1 u(1-u^{\alpha_1}-a_1w), &&x\in \Omega,t>0, \\ &\tau v_t=\Delta v-v+w^{\gamma_1}, &&x\in \Omega,t>0, \\ &w_t=\nabla\cdot(D_2(w)\nabla w)-\nabla\cdot(S_2(w)\nabla z)+\mu_2 w(1-w^{\alpha_2}-a_2u), &&x\in \Omega,t>0, \\ &\tau z_t=\Delta z-z+u^{\gamma_2}, &&x\in \Omega,t>0, \\ &\frac{\partial u}{\partial\mu}=\frac{\partial v}{\partial\mu}=\frac{\partial w}{\partial\mu}=\frac{\partial z}{\partial\mu}=0, &&x\in \partial\Omega,t>0, \\ &u(x,0)=u_0(x), \tau v(x,0)=\tau v_0(x), w(x,0)=w_0(x), \tau z(x,0)=\tau z_0(x), && x\in\Omega. \end{align*} Under appropriate regularity assumption on the initial data, the global boundedness of classical solution is obtained.