Let us assume that a GM wants to start his Celestial Empire campaign under the Qīng, on 4 July 1776. Obviously he can't use the date as such, he must convert it to the Xià calendar. This web-site will translate any Western date into its equivalent Chinese date.
I enter '4 July 1776 in Chinese calendar' in the Wolfram bar. The web-site returns the following:
19 wuyue, 4473Everything in the above answer may be used, except the year. 19 is the day. wuyue is the fifth month (as per the Wikipedia article). The Chinese didn't have any continuously numbered years until the Republic, so 4473 can't possibly be used in a Celestial Empire game. So if the GM wants to start his campaign game on 4 July 1776, the equivalent Chinese date is:
Year of the Monkey (Yang Fire)
the 19th day of the 5th month of the Year of the Fire Monkey.
This is enough to start a campaign game, as per p13 of the rules.
However, the GM and/or the players may want more accurate a reckoning for the year of the campaign game. Unless he owns a book with the correspondence of all Western years and Chinese years [I do], the average Celestial Empire GM must look up in an encyclopaedia what emperor actually ruled in 1776, when he was crowned, and deduce what year of his reign 1776 was. In our case, the emperor who ruled China in 1776 was Qiánlóng, who ascended to the throne in 1735. The year is hence the 41st year of the reign of Qiánlóng.
The full Xià date for 4 July 1776 then reads as follows:
the 19th day of the 5th month of the Year of the Fire Monkey, 41st year of the reign of Qiánlóng.