Gravitational waves provide a unique and powerful opportunity to constrain the dynamics in the interior of proto-neutron stars during core collapse supernovae. Convective motions play an important role in generating neutron stars magnetic fields, which could explain magnetar formation in the presence of fast rotation. We compute the gravitational wave emission from proto-neutron star convection and its associated dynamo, by post-processing three-dimensional MHD simulations of a model restricted to the convective zone in the anelastic approximation. We consider two different proto-neutron star structures representative of early times (with a convective layer) and late times (when the star is almost entirely convective). In the slow rotation regime, the gravitational wave emission follows a broad spectrum peaking at about three times the turnover frequency. In this regime, the inclusion of magnetic fields slightly decreases the amplitude without changing the spectrum significantly compared to a non-magnetised simulation. Fast rotation changes both the amplitude and spectrum dramatically. The amplitude is increased by a factor of up to a few thousands. The spectrum is characterized by several peaks associated to inertial modes, whose frequency scales with the rotation frequency. Using simple physical arguments, we derive scalings that reproduce quantitatively several aspects of these numerical results. We also observe an excess of low-frequency gravitational waves, which appears at the transition to a strong field dynamo characterized by a strong axisymmetric toroidal magnetic field. This signature of dynamo action could be used to constrain the dynamo efficiency in a proto-neutron star with future gravitational wave detections.