In this article, we study the stability and transition of couple stress fluid saturated porous media, heated from below and cooled from above by employing a thermal non-equilibrium model. Careful analysis shows that the thermal non-equilibrium model has a global attractor, and the global attractor only consists of the basic solution if the Rayleigh number is equal or below a threshold. In generic case where the transitioning eigenvalue has multiplicity one, we show that the transition involved is of continuous type, and the basic solution will be bifurcated to two stable convection solutions which attract globally. If the leading eigenvalue multiplicity is two, the transition is also continuous and a global attractor homeomorphic to the unit circle bifurcates. The attractor then contains four steady-state convection solutions, two of which are stable while the other two are unstable. Further numerical works give more details in the transition process including the bifurcated roll structure.