The quasi-P wave in anisotropic solids is of practical importance in obtaining maximal imaging resolution in seismic exploration. The geometrical optics term in the asymptotic expansion for the wave characterizes the high frequency part of the quasi-P wave by using two functions: a phase (traveltime) function satisfying an eikonal equation and an amplitude function satisfying a transport equation. Based on a paraxial eikonal equation satisfied by the traveltime corresponding to the quasi-P downgoing waves, two new advection equations for take-off angles provide essential ingredients for computing amplitude functions on uniform Cartesian grids. However, the radiation problem of the eikonal equation has an upwind singularity at the point source which renders all finite-difference eikonal solvers to be first-order accurate near the source. Extending an isotropic adaptive eikonal solver to the paraxial quasi-P eikonal equations can treat this singularity efficiently and yield highly accurate traveltimes and amplitudes. Numerical experiments for quasi-P traveltimes and amplitudes in transversely isotropic media with vertical symmetry axes verify that the numerical methods are efficient and accurate.