This is a tricky question to answer as it involves a great deal of speculation. Although Juliet's age is specified within Shakespeare's text when the Nurse and Lady Capulet acknowledge that she is not yet fourteen years old, we never receive specific information about Romeo's age or birth date.
Of course, something else to consider is that Shakespeare himself did not create the characters of Romeo and Juliet; they are derived from previous sources, including a poem by Arthur Brooke, which cites Romeo as having a "tender chin"--suggesting that he has not yet sprouted a beard and might be hovering around the age of puberty. Even prior to that was Bandello's novella Giulietta e Romeo, which suggests that Romeo is in his early twenties.
What we do know about Romeo is that the story Shakespeare is telling about him is set in the 14th century; thus, a best guess would be that the character Romeo was born in the 1300s. In fact, the real life lovers who inspired the story are said to have lived in Verona during 1303.
Alas, poor questioner. I know it not.
Shakespeare does not inform his readers or audience of a year for Romeo's birth. In fact, a reader never actually knows exactly how old Romeo is. We are told that Juliet is 13 in act 1 scene 2. Capulet is speaking to Paris and says "She hath not seen the change of fourteen years," meaning that Juliet has not yet turned 14.
There is speculation all over the place about Romeo's age. Some readers put him at the same age as Juliet. Most readers think he is older than Juliet by a few years. So Romeo is likely 15 or 16 years old.
The only other information that I can give you about his birth date is that it is not during the time period that Shakespeare wrote the play. The date of publication is 1597, but the play takes place during the early Renaissance. That would put Romeo being born sometime around the 14th century.