Although the SARS-CoV-2 virus has been detected in wastewater from several countries, monitoring its presence in other water matrices is still limited. This study aimed to evaluate the presence of this virus in natural and drinking water over one year of monitoring (2021). A survey of viral RNA was carried out by RT-qPCR in concentrated samples of surface water, groundwater, and drinking water from different regions of Portugal. SARS-CoV-2 RNA—quantified in genomic copies per liter (gc/L) of sampled water—was not detected in groundwater, but was detected and quantified in samples of surface water (two out of 43; 8035 and 23,757 gc/L) and in drinking water (one out of 43 samples; 7463 gc/L). The study also detected and quantified Norovirus RNA, intending to confirm the use of this enteric virus to assess variations in fecal matter throughout the sampling campaign. The samples positive for SARS-CoV-2 RNA also had the highest concentrations of Norovirus RNA—including the drinking water sample, which proved negative for fecal enteric bacteria (FIB). These results indicate that, to protect human health, it is advisable to continue monitoring these viruses, and noroviruses as fecal indicators (FI) as well—especially in low-flow water bodies that receive wastewater.