If I have an arbitrary (closed?) conducting surface and a nearby charge density, is there a simple numeric way of computing the induced charge distribution on the surface?
There is no simple way. The "standard" way is to solve Poisson equation with proper boundary conditions (constant $\varphi$ at the surface). Out of potential distribution it is easy to extract charge distribution.
For simple shapes (infinite plane, sphere, etc) it is possible to solve the problem analytically. For arbitrary shape there is no simple solution.