Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has affected millions of people globally since its first detection in late 2019. Besides humans, cats and, to some extent, dogs were shown to be susceptible to SARS-CoV-2, highlighting the need for surveillance in a One Health context. Seven veterinary clinics from regions with high incidences of coronavirus disease (COVID-19) were recruited during the early pandemic (March to July 2020) for the screening of patients. A total of 2257 oropharyngeal and nasal swab specimen from 877 dogs and 260 cats (including 18 animals from COVID-19-affected households and 92 animals with signs of respiratory disease) were analyzed for the presence of SARS-CoV-2 RNA using reverse transcriptase real-time polymerase chain reaction (RT-qPCR) targeting the viral envelope (E) and RNA dependent RNA polymerase (RdRp) genes. One oropharyngeal swab from an Italian cat, living in a COVID-19-affected household in Piedmont, tested positive in RT-qPCR (1/260; 0.38%, 95% CI: 0.01–2.1%), and SARS-CoV-2 infection of the animal was serologically confirmed six months later. One oropharyngeal swab from a dog was potentially positive (1/877; 0.1%, 95% CI: 0.002–0.63%), but the result was not confirmed in a reference laboratory. Analyses of convenience sera from 118 animals identified one dog (1/94; 1.1%; 95% CI: 0.02–5.7%) from Lombardy, but no cats (0/24), as positive for anti-SARS-CoV-2 receptor binding domain (RBD) antibodies and neutralizing activity. These findings support the hypothesis that the prevalence of SARS-CoV-2 infection in pet cat and dog populations, and hence, the risk of zoonotic transmission to veterinary staff, was low during the first wave of the pandemic, even in hotspot areas.