Random point patterns are ubiquitous in nature, and statistical models such as point processes, i.e., algorithms that generate stochastic collections of points, are commonly used to simulate and interpret them. We propose an application of quantum computing to statistical modeling by establishing a connection between point processes and Gaussian boson sampling, an algorithm for photonic quantum computers. We show that Gaussian boson sampling can be used to implement a class of point processes based on hard-to-compute matrix functions which, in general, are intractable to simulate classically. We also discuss situations where polynomial-time classical methods exist. This leads to a family of efficient quantum-inspired point processes, including a fast classical algorithm for permanental point processes. We investigate the statistical properties of point processes based on Gaussian boson sampling and reveal their defining property: like bosons that bunch together, they generate collections of points that form clusters. Finally, we analyze properties of these point processes for homogeneous and inhomogeneous state spaces, describe methods to control cluster location, and illustrate how to encode correlation matrices.