Early on in the film, Nathan suggests that he's conducting a Turing Test. However, the problem with the Turing Test (as presented in this movie) is that it doesn't really prove the machine is sentient; it only proves that the machine can fake sentience sufficiently to fool a human being.
As Caleb puts it:
CALEB: ... Like trying to test a chess computer by only playing chess.
NATHAN: How else would you test a chess computer?
CALEB: It depends what you’re testing it for. You can play it to find out
if it makes good moves. But that won’t tell you if it knows it’s playing chess. Or if it even knows what chess is.
NATHAN: So it’s simulation versus actual.
CALEB: Exactly. And I think being able to differentiate between those two is
the "Turing test" you want me to perform. The difference between an ‘AI’ and an ‘I’.
He still refers to it as a "Turing test" (probably for simplicity) but in reality, this is going well beyond the original Turing test, as described in the link you provided above. Then, later on, we have this exchange:
NATHAN: Although I’ve got to admit, I’m surprised. I mean, did we ever get
past the chess problem, as you phrased it? As in: how do you tell
if a machine is expressing a real emotion, or a just a simulated one?
NATHAN: Does Ava actually like you? Or not?
NATHAN: Though now I stop to think, there is a third option. Not whether she
does or doesn’t have the capacity to like you. But whether she’s pretending to like you.
CALEB: Why would she do that?
NATHAN: I don’t know. Perhaps - if she saw you as a means of escape.
And now we're starting to get down to Nathan's real motive in the movie. He wants to prove that Ava is truly intelligent - not just simulating intelligence at a degree that would fool most people. The means he decided on to prove that was to put Ava in captivity (a situation that any true human would find intolerable) and see if she finds the motivation and the means to escape - of her own volition, not because she was programmed to do so - by interacting with a real human and convincing him to help her.
CALEB: What was the real test?
NATHAN: You. Ava was a mouse in a mousetrap. And I gave her one way out.
[Ed: The "one way out" was her conversations with Caleb - she never gets to leave her room otherwise. Even when conversing with Caleb there's a glass wall between them, so she can't physically force him to help her.]
NATHAN: To escape, she would have to use imagination, sexuality, self-awareness, empathy, manipulation - and she did. If that isn’t AI, what the f--- is?
So it's a variation on the original Turing Test. Not for the machine to convince a person it's human, but for the machine to convince a person it deserves to be treated as a human, even though it's clearly not.