An extension of the Boussinesq-type models to weakly compressible flows is derived in the fully nonlinear case. The dispersive properties are consistent with the linear theory of compressible fluids at the long-wave limit. The particular case of a vanishing Mach number gives a quasi-incompressible model, intended for coastal wave simulations, which is a hyperbolic version of the Serre-Green-Naghdi equations, with a new treatment of the bathymetric terms. Both the compressible and quasi-incompressible models are hyperbolic four-equation models on an arbitrary bathymetry, with an exact equation of energy conservation. In addition, these models are extended to hyperbolic and fully nonlinear five-equation versions with improved dispersive properties. A remarkable property of the quasi-incompressible model with improved dispersive properties is that it is possible to decrease significantly the sound velocity, and thus the computational time, with the same accuracy or even slightly better. The numerical results show good agreement of the quasi-incompressible model with experimental data and the capability of the compressible model to calculate the decrease of tsunami velocity due to compressible effects.