Is possile change the Redirect name generated by the api?

As 3v said.

You should not be using curl, you are supposed to redirect or provide a link for the user to click on, you don’t fetch the page using curl.

Step 1 as documented here:

Is NOT an instruction to perform a curl request. you either place the https://id.twitch.tv/oauth2/authorize inside a a href:

<a href="https://id.twitch.tv/oauth2/authorize?RestOfLink">Login</a>

Or behind something like

<a href="/login/">Login</a>

And the login page does a

<?php
    header('Location: https://id.twitch.tv/oauth2/authorize?RestOfLink');

There should be no activity performed for step one of oAuth via cURL