Cook married Elizabeth Batts (1742–1835), the daughter of Samuel Batts, keeper of the Bell Inn, Wapping and one of his mentors, on 21 December 1762 at St. Margaret's Church in Barking, Essex. The couple had six children: James (1763–94), Nathaniel (1764–80, lost aboard HMS Thunderer which foundered with all hands in a hurricane in the West Indies), Elizabeth (1767–71), Joseph (1768–68), George (1772–72) and Hugh (1776–93), the last of whom died of scarlet fever while a student at Christ's College, Cambridge.