To my mind the connection between a cough and a confession is pretty straightforward.
“I can not find it in any of the dictionaries that I refer to. How often have you seen or heard it? “
I have not kept formal records on this topic. A small number, but a bit of Googlation shows it is common.
Some examples from the wild:
From Rule 34 (a novel by Charles Stross, 2004)
“He coughed to it voluntarily, and more to the point, he handed over the material”
Diary: Matthew Norman (2003)
“- posing as a friend - infiltrated her bedroom to interview the poor woman. A suspicious matron walked in, just before the camera flashed, at which point the reporter coughed to it and left.”
Hospital patient confesses to unsolved murder (2012)
“Washington, who was in prison for an unconnected offence, coughed to the crime to guard James Tomlinson.”