The context of this anecdote is the Wolfskehl Prize for the first correct proof of Fermat's Last Theorem. At its creation, the prize was endowed with the enormous sum of 100,000 German marks in gold. The Königliche Gesellschaft der Wissenschaften at Göttingen (the Göttingen Royal Society of Science), was charged with determining the eventual recipient. For a comprehensive description see the following paper:
Klaus Barner, "Paul Wolfskehl and the Wolfskehl Prize." Notices of the AMS, Vol. 44, No. 10, Nov. 1997, pp. 1294-1303.
On June 27, 1908, the day exactly eighty-nine years prior to the presentation of the prize to Wiles, the conditions for the Wolfskehl Prize endowment were laid down by the Göttingen Royal Society of Science. These were published in several journals, for instance in the Jahresbericht der DMV 17 (1908), 111–113 (with commentary by Felix Klein); the Mathematische Annalen 66 (1909), 143–144; and the Acta Mathematica 31 (1908), at the end of the volume. The prize was, according to the testament, to be valid until September 13, 2007.
A high-quality scan of the notice about the prize from Mathematische Annalen can be found at the Göttingen Digitization Center.
Unsurprisingly, the prize elicited a huge number of submissions, which were examined by "Assistenten" (literally: assistants; I do not know whether this equates to postdocs, assistant lecturers, or assistant professors) at the Mathematical Institute at Göttingen, where Edmund Landau worked from 1909 to 1934. According to the article about the Wolfskehl Prize in the German Wikipedia, which cites Paulo Ribenboim's Fermat´s Last Theorem, there were 621 submissions of supposed proofs in the first year alone. Citing an employee of the institute in the mid-1970s, there were still 3 to 4 submissions per month at that time.
A fairly extensive online search for Edmund Landau, reminiscences of people who knew him personally during the 1930s, and mathematics at Göttingen between 1909 and 1938 (the year of Landau's death) did not provide any evidence of the anecdote. The story is therefore very likely apocryphal. The earliest mention in print that I could find is this article:
Nelson Stephens, "The prize puzzle in mathematics", New Scientist, Vol. 102, No. 1409, May 10, 1984, pp. 16-19.
Referring to the Georg August University of Göttingen, the author writes:
When Edmund Landau was head of the mathematics department there from 1909 to 1934, he prepared preprinted letters beginning something along the lines of "Dear , Thank you for your manuscript on the proof of Fermat's Last Theorem. The first mistake is on line , page , and invalidates the proof."
It might be interesting to trace the history of this anecdote back through the literature; I am marking this answer as Community wiki for that purpose.