This particular anecdote about Diogenes of Sinope does indeed come from Diogenes Laertius' writing about his namesake that I discuss in the comment kindly linked to by /u/Holy_Shit_HeckHounds. As per the translation on Perseus:

Plato had defined Man as an animal, biped and featherless, and was applauded. Diogenes plucked a fowl and brought it into the lecture-room with the words, "Here is Plato's man." In consequence of which there was added to the definition, "having broad nails." To one who asked what was the proper time for lunch, he said, "If a rich man, when you will; if a poor man, when you can."

Laertius - writing several hundred years after Diogenes of Sinope lived - necessarily drew on what previous writers had said about Diogenes of Sinope. For better or worse, while Laertius mentions some sources here and there for his anecdotes about Diogenes, this particular anecdote is unsourced (but may well be from a previous now-lost book by Diodes that some have argued provides the main framework for The Eminent Lives of the Philosophers). As I say in the above answer, Laertius has a reputation for being a bit credulous, and not particularly philosophically switched on. It is likely that he had access to more than one collection of Diogenes' sayings, but it's unlikely Laertius had access to anything actually written by Diogenes (though Laertius does directly quote some of what he claims to be Diogenes' poetry, which he's likely found quoted in other sources).

Just as the Pythagoreans claimed several philosophical ideas and mathematical findings to Pythagoras, it's likely that there were plenty of funny ripostes to pretentious philosophers that got put into the mouth of Diogenes, because he was seen as the paragon of the philosophy of Cynicism (a popular philosophical position in the Hellenistic era, which isn’t just about being ‘cynical’ in the modern sense). So we should be…cynical (in the modern sense) whether Diogenes said anything he was credited with, for exactly the same reasons we should be cynical about Pythagoras coming up with the framework of personality in Plato’s Republic (which Laertius credits Pythagoras with, somewhat comically).

Of the philosophers alive during the same time as Diogenes whose voluminous writings survive, we might expect them to discuss Diogenes if he really was a consistent thorn in their side making them look pretentious. However, Plato doesn't mention him at all (despite, of course, being at the center of the anecdote here) and Aristotle seemingly mentions him once in an off-handed way. In fact, Luis E. Navia's 2005 book Diogenes the Cynic says that some have argued that Diogenes actually may not have arrived in Athens until after Plato passed away (Sinope being a Greek colony located on the modern Turkish coast, and the date of Diogenes' exile from Sinope being a source of conjecture). One suspects, if this featherless biped thing was a well-known anecdote in Plato's lifetime, that Plato would have addressed it in his writing in a way that attempts to make Diogenes look foolish.

So in all likelihood, it's a funny anecdote created by someone later to illustrate the differences between Platonism and Cynicism. As an anecdote, it's of a piece with the portrayal of Socrates in Aristophanes' The Clouds, in which Socrates - literally - has his head in the clouds, making a bunch of daft arguments. Similarly, the Cynical tradition in antiquity clearly also saw the Platonic tradition as overthinking it, barking up the wrong tree, etc - this is what the anecdote largely represents.

Mind you, Plato did mention featherless bipeds in his writings. In his dialogue The Statesman, putting this quote into the mouth of 'The Eleatic Stranger', who is conversing with the Younger Socrates (not the famous Socrates, but a younger namesake):

I say that we should have begun at first by dividing land animals into biped and quadruped; and since the winged herd, and that alone, comes out in the same class with man, should divide bipeds into those which have feathers and those which have not, and when they have been divided, and the art of the management of mankind is brought to light, the time will have come to produce our Statesman and ruler, and set him like a charioteer in his place, and hand over to him the reins of state, for that too is a vocation which belongs to him.

This quote comes as part of a larger discussion about the difficulty of coming up with useful/correct distinctions between categories - e.g., elsewhere the dialogue discusses whether 'Greek' vs 'barbarian' is a useful dichotomy, as there are many more barbarians than Greeks, and quite a lot of variation. Ultimately, The Eleatic Stranger's featherless bipeds thing is a part of a larger argument that humans are like other tame animals and do best when herded by a powerful but benevolent statesman (though whether Plato himself believes this or was presenting this more as food-for-thought/fuel-for-discussion is a bit unclear - the Eleatic Stranger isn't the character in his dialogues who usually serves as his mouthpiece). For the record, Eleatic Stranger ultimately doesn't talk about fingernails.

It is clear that Diogenes of Sinope's reputation by the time of Laertius was that he was a bit of an anarchist - clearly not the kind of person likely to be sympathetic to such arguments about benevolent statesmen (witness the likely-apocryphal stories about Diogenes meeting Alexander the Great discussed in Plutarch that Diogenes Laertius also references). Nonetheless, when Plato writes about featherless bipeds in The Statesman, the dialogue continues uninterrupted by Diogenes's comedic turn.

You may be interested in the answer to How do we know that philosophers like Socrates or Diogenes actually existed rather than just being characters fabricated by later philosophers to teach specifics schools and ideas? written by u/hillsonghoods which talks about Diogenes (albeit in less detail that Socrates).