The authors consider Galerkin and mixed finite element methods for the quasi-static thermo-poroelasticity problem with nonlinear convective transport. The problem is a nonlinear coupled problem with three subproblems, the energy balance equation for the temperature \(T\), the mass balance equation for the pressure \(p\), and the momentum balance equation for the displacement \(\textbf{u}\). A mixed element method is used for the pressure and Darcy velocity of the mass balance equation, a Galerkin finite element method is used for the temperature of the energy balance equation, and the elastic displacement of the momentum balance equation, separately. To linearize the nonlinear convective transport term in the energy conservation equation, the authors approximate \(\omega\cdot \nabla T\) as \(\omega^n \cdot \nabla T^{n+1}\), where \(\omega=-K\nabla p\) denotes the Darcy flux, \(n \geq 0\) is the time iteration index. Some suitable fully discrete finite element schemes are established. The stability is proved. Optimal error estimates are shown with an unconditional convergence. Using a comparison with a time-discrete system, these error estimates are proved without any time-step restriction. Some numerical tests are presented to check the accuracy and to show the efficiency of the proposed numerical methods.