We study semi-martingale obliquely reflected Brownian motion (SRBM) with drift in a wedge of the plane in the transient case. Our main result determines a general explicit integral expression for the Laplace transform of Green's functions of this process. To that purpose we establish a new kernel functional equation connecting Laplace transforms of Green's functions inside the wedge and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the "Basic adjoint relationship" which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the Laplace transform function in terms of contour integrals and a conformal mapping.