An assembly is provided for a turbine engine. A combustor wall of the turbine engine assembly includes a shell, a heat shield and an annular body. The annular body extends through the combustor wall. The annular body at least partially defines a quench aperture along a centerline through the combustor wall. The annular body defines a first cooling aperture fluidly coupled between a cooling cavity and the quench aperture. The cooling cavity is between the shell and the heat shield.