The short answer is “No”.
The World Health Organization is giving clear advise regarding this question:
“To date there has been no information nor evidence to suggest that the new Corona virus could be transmitted by mosquitoes.
The new Corona virus is a respiratory virus which spreads primarily through droplets generated when an infected person coughs or sneezes, or through droplets of saliva or discharge from the nose.
To protect yourself, clean your hands frequently with an alcohol-based hand rub or wash them with soap and water. Also, avoid close contact with anyone who is coughing and sneezing.”