The Prophecy
Boromir mentions Faramir in the chapter The Council of Elrond of The Fellowship of the Ring. He recounts to the Council the crucial dream of the Prophecy of Frodo and the One Ring which his brother experienced.

"On the Eve of the sudden assault [of Mordor on Osgiliath] a dream came to my brother in a troubled sleep, and afterwards a dream came oft to him again, and once to me.

In that dream I thought the eastern sky grew dark and there was a growing thunder, but in the west a pale light lingered, and out of it I heard a voice, remote but clear, crying:

Seek for the Sword that was broken
In Imladris it dwells;
There shall be counsels taken
Stronger than Morgul-spells
There shall be shown a token
That doom is near at hand
For Isildur's bane shall waken,
And the Halfing forth shall stand.

Of these words we could understand little, and we spoke to our father, Denethor, Lord of Minas Tirith, wise in the lore of Gondor." Boromir then told that Denethor had told them of Imladris. "Therefore my brother, seeing how desperate was our need, was eager to heed the dream and seek for Imladris; but since the way was full of doubt and danger, I took the journey upon myself."

Nùmenor
In the chapter The Steward and the King of The Return of the King, Faramir tells Éowyn that the brief moment of darkness that comes upon Minas Tirith "reminds me of Nùmenor... I often dream of it."

The island of Nùmenor sank when its inhabitants, the Nùmenoreans, became jealous of the elves and their immortality. Under the order of King Ar-Pharazon, who had been convinced by the deception of Sauron, they attacked the Undying Lands (Valinor) in the hope that they would win immortality if they gained control of them. They had been ordered not to sail to these Lands, and when they did, a great chasm opened in the Sea and Nùmenor was swallowed up.

The story of Nùmenor can be read in more detail in Appendix A - The Nùmenorean Kings (i) Nùmenor of The Return of the King.

The interesting thing about this event was that Tolkien had a recurring dream about it. He bequeathed it to Faramir in the chapter The Steward and the King of The Return of the King.

"For when Faramir speaks of his private vision of the Great Wave, he speaks for me. That vision and dream has been ever with me -- and has been inherited... by one of my children."

This is not surprising, as Faramir is the character Tolkien stated that he was most like.

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