Searching through online text collections turns up a lot of old uses of “quiet--too quiet” to refer to a person who is ill, possibly dying. However, using Googlebooks I turned up this, which seems to fit the modern use, from an account of the battle of Balaclava in Soldiering in Sunshine and Storm by William Douglas, 1864 (yes, I verified the date).
“The cavalry standing at their pickets, in front were the four redoubts occupied by the Turks on a range of low hills, crossing the plain from beneath the heights of the plateau to the opposite ridge near the village of Kamara. The Turks are lying lazily smoking around the guns, all so quiet—too quiet—it was only the lull before the storm, which speedily burst.”
I’m not saying this is the origin, there may well be antedates that could be found with a little more work. But it shows how far back the phrase goes, and allows us to dismiss any suggestions that it originated in movies (though of course it’s a cinematic cliche, and has been for decades).
(Found the same passage published in an 1859 magazine.)