The Gentleman's Magazine (1735) has under the title 'The joys of love never forgot: a song', a mediocre copy of verses, beginning:
'Dear Chloe, while thus beyond measure,
You treat me with doubt and disdain.;
and continuing in the third stanza:
'Old Darby, with Joan by his side,
You've often regarded with wonder:
He's dropsical, she is sore-eyed,
Yet they're never happy asunder'.
This has usually been considered the source of the names, and various conjectures have been made, both as to the author, and as to the identity of 'Darby and Joan', but with no valid results.
It is possible that the names go back to some earlier piece, and as Darby is not a common English surname, it may have originated in a real person.
There is also a well-known 19th century song of the name.
See also Michael Quinion's World Wide Words
From: He Knew He Was Right
By Anthony Trollope, Volume II, 1869
Chapter XC. Lady Rowley Conquered