"To date, no evidence has surfaced that such a conversation actually took place. The earliest known appearance of this story is in Harper's New Monthly Magazine in 1884 (the transcription above is from this source). It was repeated by M.D. Conway in his Omitted Chapters of History Disclosed in the Life and Papers of Edmund Randolph, first published in 1888. Since then, the story has appeared many times in print, usually prefaced by the phrase, "the story goes..." or something similar."
The Monticello website goes on to mention that Jefferson wrote to Lafayette advocating a bicameral government.
Also note that (1) Jefferson was in France at the time of the constitutional convention, and wouldn't have been involved in the discussion of unicameral/bicameral legislature. Although the conversation could have taken place on his return, by that point Jefferson was too busy embezzling government funds to pay authors to write articles attacking Hamilton, and (2) the story really isn't in character for Washington. Washington was quite embarassed by his lack of education and would have been very unlikely to try to go toe to toe with one of the more educated men in America at the time. It seems to me that Washington would have been far more likely to suggest that another of his friends have the conversation with Jefferson.
Update: Washington on his education. I've included one cite below. Unlike Jefferson, Adams and Hamilton, Washington did not have a college education.
There can be no doubt that Washington felt his lack of education very keenly as he came to act upon a larger sphere than as a Virginia planter. "I am sensible," he wrote a friend, of his letters, "that the narrations are just, and that truth and honesty will appear in my writings; of which, therefore, I shall not be ashamed, though criticism may censure my style." When his secretary suggested to him that he should write his own life, he replied, "In a former letter I informed you, my dear Humphreys, that if I had talents for it, I have not leisure to turn my thoughts to Commentaries. A consciousness of a defective education, and a certainty of the want of time, unfit me for such an undertaking." On being pressed by a French comrade-in-arms to pay France a visit, he declined, saying, "Remember, my good friend, that I am unacquainted with your language, that I am too far advanced in years to acquire a knowledge of it, and that, to converse through the medium of an interpreter upon common occasions, especially with the Ladies, must appear so extremely awkward, insipid, and uncouth, that I can scarce bear it in idea."
Read more: http://www.infoplease.com/t/history/true-washington/adulthood.html#ixzz2GuQ0Sd34
Here is another assertion on the same theme
The fact that Washington did not receive the same formal education that his brothers had received from England was a source of irritation and embarrassment for Washington. However, Washington eventually received a surveyor’s license from the College of William and Mary. The exact details of how long Washington attended the school or what the requirements were for the license are unclear.
The topic is also mentioned in Gordon Wood's "The Radicalism of the American Revolution", but since I'm listening to that as an audiobook, it is more difficult to cite.
My surmise that Washington is more likely to have requested that someone else initiate the conversation is developed based on Washington's work to gain approval of the constitution, described (somewhat diffusely) in "Triumvirate". Washington felt that his reputation would be diminished if he campaigned actively for the Constitution, but he wrote private letters to people he perceived as thought leaders, urging them to make their opinions known.