The present authors proposed the adjoint boundary integral method for identifying heat source or force in a domain from values of boundary integrals involving an adjoint operator. The integrals can be evaluated if boundary values are available over the entire boundary of the domain. In the present study a Poisson field source is identified from noisy and discrete boundary observations by applying the method. Adaptive Gauss quadrature enables evaluation of the boundary integrals from observations at discrete points, which deviate from the Gauss points for approximate evaluation of boundary integrals. Numerical simulations are carried out for identifying location and intensity of a concentrated source in a two-dimensional domain. Effects of errors in boundary observations, deviation of locations of observation points from the Gauss points, the order of Gauss quadrature, and the location of source on the accuracy of the identification are discussed. It is shown that the location and intensity of the source can be estimated reasonably from noisy and discrete observations by applying the adjoint boundary integral method with the adaptive Gauss quadrature.