Sherlock Holmes’ birthday is never mentioned in the canon of 60 stories by Arthur Conan Doyle. Christopher Morley, a popular early 20th-century writer and Sherlockian, proposed January 6 as Holmes' birthday.
There are a few different theories for this date. In The Valley of Fear (
bit.ly/2L8ImLZ), Holmes is a little grumpy and irritable when the story begins on January 7, perhaps suggesting he had too much to drink the night before (on his birthday). Another theory: Holmes quotes “The Twelfth Night" more then any other Shakespeare play, and the twelfth night after Christmas happens to be January 6.
As to the year? Conan Doyle made this clearer. In His Last Bow (
bit.ly/3beyew7), which is set in 1914, he mentions Holmes is 60 years old, which would put his year of birth at 1854.
The first meeting of the Baker Street Irregulars (the first Sherlock Holmes society) was January 6, 1934.