Early sources such as this 1896 biography cite "Istorič. Věstník 1883 p. 376" by one Prof. M. Berg. as their source for this anecdote, but they don't specify that Flaubert made the remark, and I haven't been able to track Berg's book down.
The earliest source I could find, Květy (1891), says the following, translated from pages 617/18:
Prof. M. Berg (Istorič. Věstník 1883 p. 376) says that in 1878 a good acquaintance of his asked Turgenev in Paris if one could find in France all that an educated Russian needed, if he did not wish to be in Russia for some time.
Turgenev answered something like this: [...] "Russia and Russians are something quite special, for no one understands us properly, least of all the French. I live here in the circles of a higher intelligence, which cannot see beyond its nose. The French do not understand what is beautiful and brilliant about other peoples. There are almost no English, German, or Wallachian spirits for the French. I'm not even talking about us [Russians specifically] anymore ... Only what's French is perfect for them. I once went to explain to one of my friends—he was a very sensible and sharp-eyed Frenchman—about the beauty of one of Pushkin's poems, which in my opinion is a wonderful pearl of poetry and perfect in every respect. The Frenchman heard me and said: 'c'est plat, mon cher!' (That's crude, my dear.)"
Another early source is Two Russian Reformers: Ivan Turgenev, Leo Tolstoy (1911), which appears to be translating from the same sources:
So far as his own work was concerned Turgenev had no belief in the expressions of French admiration. Nor did he believe that they appreciated the national genius of his country. "The French," he wrote once, "recognise no originality whatever in other peoples. The genius of England, of Germany, of Italy, is a dead letter or almost a dead letter to them; as for my own country, do not let us speak of it! Apart from their own affairs, they are interested in nothing, they know nothing." [...] And he quotes, as though once and for ever to sum up the French standpoint towards Russian literature, the comment of a very distinguished Frenchman upon one of the masterpieces of Pushkin: "C'est plat, mon cher!"
Thus here, too, all we learn for certain is that the comment was made by a "distinguished Frenchman."
Turgenev was certainly on close terms with the French, and seemingly most of all with Flaubert; one early source notes that he was "intimate with French literary circles—with Mérimée, Flaubert, and the young naturalists." Per volume 116 of The Fortnightly, Flaubert and Turgenev were very close. Apart from Turgenev, Flaubert said, "I do not know a human being with whom I can talk over things which I have really at heart"; Turgenev, for his part, went to Paris to meet "Flaubert, whom I love much." It's not unimaginable that Flaubert was the dismissive Frenchman.