The light trajectory in the gravitational field of one body at rest with monopole and quadrupole structure is determined in the second post-Newtonian (2PN) approximation. The terms in the geodesic equation for light rays are separated into time-independent tensorial coefficients and four kinds of time-dependent scalar functions. Accordingly, the first and second integration of the geodesic equation can be reduced in each case to only four kinds of scalar master integrals. These integrals can be solved in closed form by recurrence relations. The 2PN terms of monopole and quadrupole contribute less than 1 nanoarcsecond to the total light deflection. There are, however, enhanced terms in the 2PN light deflection, both in the case of monopole and quadrupole. These enhanced 2PN terms are caused by the use of an impact vector which is indispensable for modeling of real astrometric measurements. In the case of grazing light rays at Jupiter and Saturn, the enhanced 2PN terms, caused by the quadrupole structure of the massive body, amount up to 0.95 microarcseconds and 0.29 microarcseconds, respectively. Thus, the 2PN quadrupole terms are relevant for high-precision astrometry on the submicroarcsecond scale of accuracy.