In this work we carry out an exploratory analysis of online conversations on the Italian Facebook during the recent COVID-19 pandemic. We analyze the circulation of controversial topics associated with the origin of the virus, which involve popular targets of misinformation, such as migrants and 5G technology. We collected over 1.5 M posts in Italian language and related to COVID-19, shared by nearly 80k public pages and groups for a period of four months since January 2020. Overall, we find that potentially harmful content shared by unreliable sources is substantially negligible compared to traditional news websites, and that discussions over controversial topics has a limited engagement w.r.t to the pandemic in general. Besides, we highlight a "small-worldness" effect in the URL sharing diffusion network, indicating that users navigating through a limited set of pages could reach almost the entire pool of shared content related to the pandemic, thus being easily exposed to harmful propaganda as well as to verified information on the virus.