What I Know and What I Don't
I've spent enough time digging through niche software, obscure tools, and weird corner-of-the-internet products to say this straight: I cannot find a reliable, verifiable reference to something called Go Away Big Green Monster in any legitimate source. No official website, no documentation, no download page that isn't a sketchy ad farm or an empty landing page. That means I can't honestly give you a how-to guide, a tutorial, or a download link for it. I won't make one up just to fill space, and I won't send you to a site I haven't verified because that would be doing you a disservice.
Go Away Big Green Monster — what people usually mean
There are a couple of things this phrase sometimes gets attached to, and they're all different: A children's book concept. "Go Away, Big Green Monster!" is a well-known interactive picture book by Ed Emberley about a kid imagining a monster and making it go away by turning its body parts on and off. It's been adapted into apps and digital versions over the years. If that's what you're looking for, it's available as a book, and there have been app versions at various points — though the app marketplace versions tend to change names, get pulled, or become paid again. The book itself is straightforward. A browser extension or ad-blocker nickname. On some forums, people have used phrases like "big green monster" informally to refer to something blocking ads or tracking scripts, usually in the context of uBlock Origin or similar tools. I've never seen that as a formal name for any software, and I'd treat any site claiming that as suspicious.
A scareware or fake antivirus product. This is the category I want to flag most directly. There have been countless instances of malicious websites and rogue pop-ups using monstrous, dramatic names to scare users into downloading "cleaners" or "security tools" that are themselves harmful. If you saw this name advertised somewhere, especially on a site full of countdown timers or red warning boxes, do not download anything from it.
Get the Full Details

What I'd do if I were in your shoes
I'd go back to wherever you ran into this term and check the URL. If it was a random search result, a YouTube comment, or a pop-up on a site, the likelihood that it's legitimate is very low. If it was mentioned in a reputable children's media context, it's almost certainly the Ed Emberley book and related family-friendly apps. If you can tell me where you encountered the name — a forum thread, an app store listing, a website — I can help you figure out what it actually is or whether it's safe. Without that context, the honest answer is that I don't know what you're looking for, and I'm not going to guess.