Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the learning model with boundary information from scientific knowledge. However, despite recent work on boundary-integrated GPs, such models largely cannot accommodate boundary information on irregular (i.e., non-hypercube) domains, and do not provide sample path smoothness control or approximation error analysis, both of which are important for reliable surrogate modeling. We thus propose a novel BdryMatérn GP modeling framework, which can reliably integrate Dirichlet, Neumann and Robin boundaries on an irregular connected domain with a boundary set that is twice-differentiable almost everywhere. Our model leverages a new BdryMatérn covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with Matérn kernel, we prove that sample paths from the BdryMatérn GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMatérn kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMatérn GP in a suite of numerical experiments on incorporating broad boundaries on irregular domains.