A strong candidate for the early propagation—if not the origin itself—of this usage is the song "It's Magic," sung by Doris Day in the 1948 film Romance on the High Seas. The lyrics begin as follows:
You sigh, a song begins / You speak and I hear violins / It's magic / The stars desert the skies / And rush to nestle in your eyes / It's magic
Here's a clip of Ms Day's rendition of the song from the movie in question.
An Elephind newspaper database search turns up a couple of references to this song, many years apart. In "Torchy," in the [Golden] Colorado Transcript (September 30, 1948), someone uses the lyrics as the lead-in to a description of Harley Johnson, the Senior of the Week at Golden High School:
You sigh, the song begins, you speak, and I hear violins—“It’s Magic.” The stars desert the skies and rush to nestle in your eyes—“lt’s Magic.” Junior, or Long John (which ever you prefer) was born November 7, 1932 in Denver, Colo. He is six-feet two-inches tall and has red hair and freckles. Harley plays safe by liking all the girls but by the same token plays a good end by leaving them all alone.
In a Valentine's Day personal ad posted in the [Kent, Ohio] Daily Kent Stater (February 14, 1989), it segues into a jazz riff:
"You speak and I hear Violins, It’s magic." Well, actually I hear jazz trumpets--I Love You Noe! Sue
The third and last instance of "I hear violins" in the Elephind search results doesn't refer to "It's Magic" and, indeed, suggests that the violins, if heard, would signal not dreamy love but weepy nostalgia. From Steve Pletkin, "Nice Predictions; Awards, Good and Bad; Final Column Ever!" in the Santa Rosa [California] Junior College Oak Leaf (January 10, 1991):
Oh well, this is it, my last issue as both sports editor and editor-in-chief. It was great while it lasted, but like all good things it must come to an end. Did I hear violins? Nah!
What these instances have in common is the idea that "I hear violins" means "I imagine an appropriate musical score to accompany the intense emotions I am feeling."