A three-dimensional quasi-dynamic finite element approach is presented for the simulation of fluid-driven seismicity on a fault with rate-and-state friction. The coupled nonlinear hydro-mechanical equations of the fault and surrounding matrix are solved monolithically using the Imperial College Geo-mechanics Tool (ICGT) to obtain the fluid pressure and displacement fields. This study focuses on advances in the friction module of ICGT, where the augmented Lagrangian method is used to apply the contact constraints of the fault surface to the finite element model, which utilizes the advantages of both penalty and Lagrange multiplier methods. The stick-predictor slip-corrector algorithm proposed by Hosseini et al. for the rate-and-state friction law is developed to get better convergence. Our numerical approach is used to study the dynamic response of a fault to fluid injection, which is represented as an explicit surface in the model using zero-thickness interface elements. To generate an equivalent effect of radiation damping and prevent unbounded slip rates in a quasi-dynamic framework, a velocity-dependent cohesion term is introduced to the shear stress of the fault. It is shown that the spatial mesh size and temporal time step in such problems must also meet specific criteria to guarantee the convergence of an iterative Newton-Raphson solver. Changes in pore pressure trigger an aseismic slip front that propagates along the fault, leading to failure of the seismogenic areas. All phases of a seismic cycle for a fault, including the transition between the stick and slip states, are captured by this method. Normal and shear stresses at the slip state of the fault are coupled to each other and can generate complex stress paths, which are not readily captured using available uncoupled frameworks.