This dialogue from the film borrows from a dream Frodo has in Bombadil's house:
That night they heard no noises. But either in his dreams or out of them, he could not tell which, Frodo heard a sweet singing running in his mind; a song that seemed to come like a pale light behind a grey rain-curtain, and growing stronger to turn the veil all to glass and silver, until at last it was rolled back, and a far green country opened before him under a swift sunrise.
Fellowship of the Ring Book I Chapter 8: "Fog on the Barrow-Downs"
This is referenced again at the end of Return of the King, when Frodo departs on the Elven-ship:
[T]he ship went out into the High Sea and passed on into the West, until at last on a night of rain Frodo smelled a sweet fragrance on the air and heard the sound of singing that came over the water. And then it seemed to him that as in his dream in the house of Bombadil, the grey rain-curtain turned all to silver glass and was rolled back, and he beheld white shores and beyond them a far green country under a swift sunrise.
Return of the King Book VI Chapter 9: "The Grey Havens"
In Letter 91, Tolkien links this dream explicitly to Frodo going "over the Sea":
Frodo will join [Bilbo and Elrond and Galadriel] and pass over the Sea (linking with the vision he had of a far green country in the house of Tom Bombadil).
The Letters of J.R.R. Tolkien 91: To Christopher Tolkien. November 1944
There's some debate over whether this is specifically a reference to Valinor or to Tol Eressëa, but regardless; it is unquestionably referring to the Undying Lands in some capacity.