http://fallout.wikia.com/wiki/The_Replacement

Where do I get this quest? There's no info on it in the wiki other than that it's an Institute side quest. It doesn't even say who is supposed to give me it.

Is this even a real thing?

It ~may~ be a random event "quest" that can be triggered at any number of "trigger points".... if it's the one I think it is, anyway, you come across two identical NPCs, one holding a weapon on the other and each claiming that the other is a synth that is trying to replace him. You can talk to them both and decide who you believe, which one is the synth, and which one you want to help / kill off....

I just got to the Institute myself, so if it is an actual quest ~from~ them, I haven't got to it yet.

the quest is cut content, it does not appear in the final game at all.

I experienced this last week in a town. Two dudes named "Art" had guns pointed at each other, shooting and shouting at each other, and they looked exactly the same. All I did was talk to them -- no one was killed. I didn't think much of it at the time that one could be a synth since I wasn't even halfway through the game. I wish I could see it again...