The O'Brien family was the family of Miles and Keiko O'Brien.
The two were married on October 24, 2367, after Data introduced them aboard the USS Enterprise-D. (TNG: "Data's Day")
Their first child, Molly, was born in 2368. (TNG: "Disaster").
After the O'Briens moved to Deep Space 9, their second child and first son, Kirayoshi, was born in early 2373 after being carried to term by Kira Nerys. (DS9: "Body Parts", "The Begotten").