In this article, we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary-scale atmospheric and oceanic flows. The model is governed by the two-dimensional, quasi-geostrophic equation for the conservation of vorticity in an east-west oriented, periodic channel. The main result is the existence of Hopf bifurcation of the flow as the Reynolds number crosses a critical value. The key idea in proving this result is translating the eigenvalue problem into a difference equation and treating the latter by continued-fraction methods. Numerical results are obtained by using a finite-difference scheme with high spatial resolution and these results agree closely with the theoretical predictions. The spatio-temporal structure of the limit cycle corresponds to a wave that propagates slowly westward and is symmetric about the midaxis of the channel. For plausible paramater values that correspond to midlatitude atmospheric flows, the period of this wave is 20--25 days.