The ten-moment equations are considered as a first-order alternative of Navier-Stokes equations when the effect of heat transfer is negligible. This model takes the form of first-order hyperbolic conservation laws, which carry many numerical advantages. However, the applicability of this model is still limited due to the lack of appropriate turbulence models. Applying the Reynolds-averaging concept to the ten-moment model, a set of governing equations for turbulent flow can be obtained, which is referred to as the Reynolds-averaged ten-moment equations. The traditional turbulence models designed for the Reynolds-averaged Navier-Stokes (RANS) equations are not ideal for the Reynolds-averaged ten-moment equations, as the extra partial differential equations (PDEs) introduce second-order derivatives. These terms destroy the pure hyperbolic nature of the original system of equations, which consequently removes all numerical advantages of first-order systems. To maintain the first-order hyperbolic form, the desired turbulence model should remain in the same form. Recently, a hyperbolic-relaxation turbulence model has been proposed by the authors, which is developed by hyperbolizing Prandtl's one-equation model using a relaxation method known as the Chen-Levermore-Liu p-system. Unfortunately, developing a hyperbolic version of two-equation models using the same method is very difficult. This is because the diffusion coefficients of the two-equation models are more complicated than in the one-equation model. In this paper, another relaxation method, the Cattaneo-Vernotte approach, is used to develop the hyperbolic-relaxation form of classical two-equation models. The solution of the resulting equations exhibits dispersive wave behaviour. To study this feature, a dispersion analysis of the Reynolds-averaged ten-moment equations with the new turbulence models is presented. Several numerical experiments are studied to investigate the effect of the relaxation parameters. The derived turbulence models are then coupled to the Reynolds-averaged ten-moment equation and further validated by solving a canonical two-dimensional turbulent plane mixing-layer problem, planar free-jet problem, and circular free-jet problem.