That quote is clearly not from Socrates, or even claimed to have been from Socrates by any ancient author, as the source you link to notes.
However, it does show up frequently, and my view of it has generally been that it could have originated as a loose translation of the Superior/Right Argument character from Aristophanes' Clouds, who does make similar claims:
>And so, young man, that’s why you should choose me, the Better Argument. Be resolute. You’ll find out how to hate the market place, to shun the public baths, to feel ashamed of shameful things, to fire up your heart when someone mocks you, to give up your chair when older men come near, not to insult your parents, nor act in any other way which brings disgrace or which could mutilate your image as an honourable man.
This is very problematic in that, first, Clouds is a comedy mocking Socrates, second, Socrates' views at that point in the play are supposed to be represented by the Inferior/Wrong Argument character, not the Superior Argument character, and third, Aristophanes is mocking the claims of both characters.
In a fashion typical of Aristophanes, the Inferior Argument wins the debate with a joke about the audience's sexual proclivities that convinces the Superior Argument that its cause is hopeless.