The phrase to talk a glass eye to sleep is applied either to a wearisome talker, or, on the contrary, to a persuasive talker.

It is in the latter sense that the first two occurrences of the phrase that I have found are used—in connection with boxing.

1-: The first of these occurrences is from Jack O’Brien’s Sport Round-Up, published in the Daily Telegraph (Sydney, New South Wales) of Monday 29^th December 1952:

Boxing followers say that Carruthers’ ¹ trainer Bill McConnell ² is entitled to the Oscar as the most convincing talker of the year.
Bill did a fine job of publicity for Carruthers in South Africa, and talked nearly a stone off himself while he was away.
South African sportswriters were so impressed with his loquacity that they named him “Silent Bill.”
But Bill says that American featherweight boxer Johnny O’Brien ³, who was here early this year, could play him on a break.
“Johnny was a beaut,” said Bill.
“He could talk a glass-eye to sleep.”

¹ James Carruthers (1929-1990) was an Australian boxer.
² William McConnell (1906-1970) was an Australian boxing trainer.
³ Johnny O’Brien (born 1933) is a U.S. former boxer.

2-: The second occurrence of the phrase that I have found is from Fighting Facts, by Tommy Farr, published in the Sunday Pictorial (London, England) of Sunday 8^th August 1954:

Don Cockell ⁴ and manager John Simpson have sailed from New York for home.
I hope that before they left John put the negotiations for a Cockell-Rocky Marciano ⁵ world title fight into the hands of Jack Hurley ⁶, Harry Matthews’s manager.
I once said that Hurley could talk a glass eye to sleep. He could be the ideal man to convince the New York boxing big-wigs that Cockell deserves to fight Marciano next June.

⁴ Donald Cockell (1928-1983) was an English boxer.
⁵ Rocky Marciano (Rocco Francis Marchegiano – 1923-1969) was a U.S. boxer.
⁶ Jack Hurley (1897-1972) was a U.S. boxing manager and trainer.

3-: The next-earliest occurrence of the phrase that I have found is from an interview of Sarah Churchill ⁷, by Judy Bachrach, originally published in The Washington Post (Washington, D.C.) and reprinted in several U.S. newspapers in March 1976—for example in The Advocate (Newark, Ohio) of Thursday 11^th March 1976; the following passage evokes an incident that occurred after the death of her second husband, Anthony Beauchamp:

It was five months after that, that she was arrested in Malibu on a drunk charge, having ostensibly used “unladylike language” to the phone operators and the sheriff’s deputies.
“When I’m in trouble,” Sarah Churchill explains, “I don’t want any close friends around me. I don’t want to impinge. When I can’t contribute to life and if I have nothing to give, I’d rather go away in a corner and look at the sea. But apparently I got on the telephone one night.
“I was talking to a very good friend in L.A., a poet — we used to read our poetry to each other. I have a Welsh housekeeper who says I could talk a glass eye to sleep, and I talked him to sleep. Somehow the operator got on. Only when he wakes up in the morning, I’m in jail.”

⁷ The daughter of Winston Churchill (1874-1965), Sarah Churchill (1914-1982) was an English actress, author and painter.

4-: The phrase then occurs in an advertisement for Smith Knight Fay, car dealers, published in The Guardian (London and Manchester, England) of Monday 2^nd August 1982:

Our password since we started in the business has been Personal Service and you’ll find a phone call well worth your while (a far cry indeed from the usual slick talkers who can talk a glass eye to sleep!)

5-: A variant of the phrase occurs in a quiz published in the Liverpool Echo (Liverpool, Merseyside, England) of Thursday 26^th October 1995:

Which late British comedian said: “I’m not saying I’m boring but I once sent a glass eye to sleep”?
[Answer: Leslie Dawson (1931-1993)]

6-: Another variant occurs in The Age (Melbourne, Victoria) of Wednesday 11^th August 1999:

Who said that?
“With all due respect to Eddie … he could put a glass eye to sleep.”
— Veteran English snooker player Willie Thorne ⁸ on Australian legend Eddie Charlton ⁹, while singing the praises of Charlton’s heir apparent, Quinten Hann ¹⁰.

⁸ William Thorne (born 1954) is an English former snooker player.
⁹ Edward Charlton (1929-2004) was an Australian snooker and billiards player.
¹⁰ Quinten Hann (born 1977) is an Australian former snooker player.

7-: The following is from an article by Anne Davies about the appointment of the Australian businessman and media executive Jonathan Shier (born 1947) as managing director of the Australian Broadcasting Corporation—article published in The Sydney Morning Herald (Sydney, New South Wales) of Saturday 21^st October 2000:

Shier presented well to the board.
“He could talk a glass eye to sleep,” quipped one former broadcaster of Shier’s considerable verbal skills.