Understanding the formation and evolution of low surface brightness galaxies (LSBGs) is critical for explaining their wide-ranging properties. However, studies of LSBGs in deep photometric surveys are often hindered by a lack of distance estimates. In this work, we present a new catalogue of 479 LSBGs, identified in deep optical imaging data from the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP). These galaxies are found across a range of environments, from the field to groups. Many are likely to be ultra-diffuse galaxies (UDGs). We see clear evidence for a bimodal population in colour-Sersic index space, and split our sample into red and blue LSBG populations. We estimate environmental densities for a sub-sample of 215 sources by statistically associating them with nearby spectroscopic galaxies from the overlapping GAMA spectroscopic survey. We find that the blue LSBGs are statistically consistent with being spatially randomized with respect to local spectroscopic galaxies, implying they exist predominantly in low-density environments. However, the red LSBG population is significantly spatially correlated with local structure. We find that 26+/-5 per cent of isolated, local LSBGs belong to the red population, which we interpret as quiescent. This indicates that high environmental density plays a dominant, but not exclusive, role in producing quiescent LSBGs. Our analysis method may prove to be very useful, given the large samples of LSB galaxies without distance information expected from e.g. the Vera C. Rubin observatory (aka LSST), especially in combination with upcoming comprehensive wide-field spectroscopic surveys.