No, I'm pretty sure it came from hip hop culture, like a lot of slang recently. I'm basing this purely my anecdotal observation of the kind of people who use it most frequently.

I've heard it in lyrics from long before twitch existed

In Black slang, to cap about something is “to brag,” “to exaggerate,” or “to lie” about it. This meaning of cap dates back to the early 1900s.

https://www.dictionary.com/e/slang/no-cap/

Green's Dictionary of Slang - Cap

This was my theory too when I first heard it said. It wasn’t till later when I saw it spelled that I realized it’s “no cap” and not “no kap”.

I still got perfect understanding of the meaning from thinking about it in terms of the Kappa emote.

I'm going to go with this theory, because I at least know what Kappa is