In this work, we address the implementation and performance of inexact Newton-Krylov and quasi-Newton algorithms, more specifically the BFGS method, for the solution of nonlinear partial differential equations (PDEs) models arising in cardiac electrophysiology and mechanics, and compare them to standard Newton-Krylov methods. This is done through a systematic analysis of the performance of the solvers with respect to the problem size, the magnitude of the data and the number of processors. In case of cardiac electrophysiology, we consider as a model problem the Bidomain system, consisting of two parabolic reaction-diffusion PDEs modeling the spread of electrical impulse in the cardiac muscle. In case of cardiac mechanics, we consider as a model problem the equations of finite elasticity in three-dimensions, taking into account the anisotropy and almost-incompressibility of the cardiac tissue. The results of parallel numerical simulations performed on Linux clusters suggest that, for both models, quasi-Newton methods yield the best computational performance. This work was part of the MICROCARD project. This project has received funding from the European High-Performance Computing Joint Undertaking EuroHPC (JU) under grant agreement No 955495. The JU receives support from the European Union's Horizon 2020 research and innovation programme and France, Italy, Germany, Austria, Norway, Switzerland.