Monitoring the quality of water is a requisite to prevent outbreaks related to waterborne diseases, predominantly caused by pathogens like enteric viruses, usually transmitted via the fecal-oral route. This study aimed to survey a group of enteric viruses (Enterovirus, Norovirus genogroups I and II, and hepatitis A virus) in two surface water sources of drinking water, also intending to evaluate the extent of their elimination in the two water treatment plants (WTPs) involved in drinking water production. Correlations between these viruses and fecal indicator bacteria (FIB) were also evaluated. Positive samples for viral RNA were recurrently found by reverse transcription quantitative PCR (RT-qPCR) and quantified, in genomic copies per liter (gc/L) of sampled water. Viral RNAs were detected in 14 out of 27 samples of surface water, and 21 out of 36 samples of drinking water, NoV II having been the most frequently detected in both (0–78.6 gc/L and 0–12.5 gc/L, respectively). Both WTPs showed variable efficacies in the elimination of viral RNA. Only one correlation was found with FIB, between NoV II and intestinal enterococci. These results recommend the monitoring of enteric viruses over time and their inclusion in the mandatory analysis of water quality.