In Star Trek: First Contact, Zefram Cochrane makes his first warp flight in 2063.
Cochrane pilots the Phoenix, humanity’s first warp-capable spacecraft, from a missile complex near Bozeman, Montana. The flight takes place on April 5, 2063, shortly after Earth has been devastated by World War III. Its warp signature attracts the attention of a passing Vulcan survey ship, the T’Plana-Hath, whose crew lands and makes peaceful contact with Cochrane and his team.
That encounter becomes humanity’s first official contact with an extraterrestrial civilization and eventually leads to the formation of the Federation. In the film’s time-travel story, the crew of the USS Enterprise-E must protect the Phoenix’s launch because the Borg are attempting to prevent humanity from reaching warp speed.
A common mix-up is confusing 2063 with 2073, the year associated with a quotation attributed to Cochrane. The first flight itself is firmly dated to 2063.