How to check the birth time from the moon constellation?

Because the moon sign is calculated according to the lunar calendar, according to the information you provided, your moon sign is Virgo, that is, your lunar birth date is: 1987 1.02.

but .......

If February 9th is your birthday, your moon sign should be Gemini or Cancer.

If you were born in the morning, you are a Gemini, that is, the time is about February 9 13:30.

But if you come out in the afternoon, your moon sign is Cancer. ...