All good Star Wars fans know that the 4th of May is Star Wars day, so my question is: Is there an equivalent day for Star Trek? If so, when is it?
"Star Trek Day" is the 8th of September.
The first episode of The Original Series aired on 8 September 1966. The graphic above is from an advertisement that Oreo Cookies made to celebrate Star Trek Day, with approval from CBS (which owns the Star Trek television series).
Google has also traditionally done its special Star Trek Google Doodles on the 8th of September:
Many Star Trek web sites / fan sites / fanzines agree, treating the 8th of September as a special day (one "worthy of song", as the Klingons might say).
- TrekNews.net — Star Trek Celebrates 49 Years
- Star Trek New Voyages — September 8, 2015: Happy 49th Birthday Star Trek!
Furthermore, the UFStarfleet group (one of the major Star Trek fan groups) maintains that September 8 is "Star Trek Day":
EVENT: Star Trek Day
DATE: September 8
DESCRIPTION: Celebration commemorating the premiere of Star Trek on television in 1966. Celebrated across the Internet.
Also, here, Trek Movie declares September 8 to be Star Trek Day.
Also, this Facebook group:
Of course, 8 September 2016 will be a particularly important day of celebration for Star Trek fans worldwide, given that it is will be the 50th anniversary of the franchise.
Interestingly, Star Trek was first seen in Canada, on the 6th of September — two days before the US got to see it, as the Star Trek New Voyages article points out.
There is also "First Contact Day", which is the 5th of April, but this is more in-universe than out-of-universe (and I interpreted the question as being out-of-universe).
In universe, one of the most celebrated days for humans is April 5 "First Contact Day" celebrated to honor both Zefram Cochrane's first warp 1 flight of the Phoenix and first open contact between Humans and Vulcans on that date in 2063 in Bozeman, Montana.
This date has been adopted by many fans and is the closest equivalent to Star Wars' May the 4th.
EDIT to provide Doc with requested evidence: