Quantification of waterborne pathogens in water sources is essential for alerting the community about health hazards. This study determined the presence of human enteric viruses and protozoa in the Bagmati River, Nepal, and detected fecal indicator bacteria (total coliforms, Escherichia coli, and Enterococcus spp.), human-fecal markers (human Bacteroidales and JC and BK polyomaviruses), and index viruses (tobacco mosaic virus and pepper mild mottle virus). During a one-year period between October 2015 and September 2016, a total of 18 surface water samples were collected periodically from three sites along the river. Using quantitative polymerase chain reaction, all eight types of human enteric viruses tested—including adenoviruses, noroviruses, and enteroviruses, were detected frequently at the midstream and downstream sites, with concentrations of 4.4–8.3 log copies/L. Enteroviruses and saliviruses were the most frequently detected enteric viruses, which were present in 72% (13/18) of the tested samples. Giardia spp. were detected by fluorescence microscopy in 78% (14/18) of the samples, with a lower detection ratio at the upstream site. Cryptosporidium spp. were detected only at the midstream and downstream sites, with a positive ratio of 39% (7/18). The high concentrations of enteric viruses suggest that the midstream and downstream regions are heavily contaminated with human feces and that there are alarming possibilities of waterborne diseases. The concentrations of enteric viruses were significantly higher in the dry season than the wet season (p < 0.05). There was a significant positive correlation between the concentrations of human enteric viruses and the tested indicators for the presence of pathogens (IPP) (p < 0.05), suggesting that these IPP can be used to estimate the presence of enteric viruses in the Bagmati River water.