I assume that imranal asks how to find numerically a geodesic connecting two given points if the connection is given. One way to do it is to implement the solution of the ODE system he wrote in his question numerically (there are many effective ways for it) and then use this implimintation to find the correct initial velocity vector $(u', v')$ such that the geodesic starting from $(u, v)$ in the direction $(u', v')$ hits the second point. One can slighly improve this search but the principle remains the same.
Alternative method which actually needs that the connection you have comes from a Riemannian metric would be to use the curve shortening flow: start with any curve connecting these two points (say, a straight line) and then start to deform the curve such that the velocity vector of the deformation is orthogonal to the velocity vector of the curve and its length is the geodesic curvature times some fixed function which vanishes at the first and last points of the curve. This deformation should normally converge to a geodesic.
I expect that there should be effective implimentation of this idea as a numerical scheme but I am not an expert in these questions.
Of course geodesic does not always exist (as explained by Igor) and may be is not unique (as explained by Narasimham) or/and the second procedure may converge to a geodesic which is not a minimal one.
