The company social security account opening time is June. Can I change it to May?

You can't. According to the provisions of the Social Insurance Law, the company's social security account opening time is kept in the social security information database terminal, and the account opening time is based on the time of paying social endowment insurance, and June cannot be changed to May. Social insurance is a social and economic system that provides income or compensation for those who lose their ability to work, are temporarily unemployed or suffer losses due to health reasons.