No we don’t know for sure. The odds are one-in-365 that it’s right. The only clue as to the time of year is found in Luke.
And there were shepherds living out in the fields nearby, keeping watch over their flocks at night. Luke 2:8

Israel experiences cold winters (sometimes snow) in December and traditionally shepherds are in their fields from Spring to Fall. No way would they be out in the cold at night. Besides, the fields would be sparse with food.

I found this information online:

“Some sources report that Theophilus of Antioch (ca. 171-183) was the first to identify December 25 as the birth date of Christ. Others say that Hippolytus (ca. 170-236) was the first to claim that Jesus was born on December 25. A strong theory suggests that this date was eventually chosen by the church because it aligned closely with a major pagan festival, dies natalis solis invicti (birth of the invincible sun god, Saturnalia), therefore, allowing the church to claim a new celebration for Christianity.

Ultimately, December 25 was chosen, perhaps as early as 273 AD. By 336 AD the Roman church calender definitively records a nativity celebration by Western Christians on this date.”The term Christmas appeared in Old English as early as 1038 AD as Cristes Maesse, and later as Cristes-messe in 1131 AD. It means “the Mass of Christ.” This name was established by the Christian church to disconnect the holiday and its customs from its pagan origins.”

The date is not important. The meaning is. Christians use December 25 to celebrate God’s coming to earth in the form of a child. Since God is eternal and has no birthday, picking December 25th should not make a difference.