Solid-state single-photon emitters (SPE) are a basis for emerging technologies such as quantum communication and quantum sensing. SPE based on fluorescent point defects are ubiquitous in semiconductors and insulators, and new systems with desirable properties for quantum information science may exist amongst the vast number of unexplored defects. However, the characterization of new SPE typically relies on time-consuming techniques for identifying point sources by eye in photoluminescence (PL) images. This manual strategy is a bottleneck for discovering new SPE, motivating a more efficient method for characterizing emitters in PL images. Here we present a quantitative method using image analysis and regression fitting to automatically identify Gaussian emitters in PL images and classify them according to their stability, shape, and intensity relative to the background. We demonstrate efficient emitter classification for SPE in nanodiamond arrays and hexagonal boron nitride flakes. Adaptive criteria detect SPE in both samples despite variation in emitter intensity, stability, and background features. The detection criteria can be tuned for specific material systems and experimental setups to accommodate the diverse properties of SPE.