We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned to never be absorbed... The technique relies on a coupling procedure that is related to Doeblin's type conditions. The main novelty is that we modulate each coupling step depending both on a final horizon of time --for survival-- and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment. (read more)