No. According to the World Health Organization, there is no evidence to suggest this new virus can be transmitted by mosquitoes. COVID-19 is a respuratory virus which spreads primarily through droplets generated when an infected person shares saliva or discharge from the nose, such as coughs or sneezes. Mosquitoes transmit differents diseases - such as West Nile virus (WNV).
Mosquitoes become infected when they feed on infected birds, which circulate the virus in their blood for a few days. The virus eventually gets into the mosquito's salivary glands. During later blood meals (when mosquitoes bite), the virus may be injected into humans and animals, where it can multiply and possibly cause illness.