Weak gravitational lensing by galaxy clusters on faint higher redshift galaxies has been traditionally used to study the cluster mass distribution and as a tool to identify clusters as peaks in the shear maps. However, it becomes soon clear that peaks statistics can also be used as a way to constrain the underlying cosmological model due to its dependence on both the cosmic expansion rate and the growth rate of structures. This feature makes peak statistics particularly interesting from the point of view of discriminating between General Relativity and modified gravity. Here we consider a general class of f(R) theories and compute the observable mass function based on the aperture mass statistics. We complement our theoretical analysis with a Fisher matrix forecast of the constraints that an Euclid - like survey can impose on the f(R) model parameters. We show that peak statistics alone can in principle discriminate between General Relativity and f(R) models and strongly constrain the f(R) parameters that are sensitive to the non - linear growth of structure. However, we also find a degeneracy between f(R) and dark energy models and the adopted relation between cluster mass and concentration.