I've never heard it used that way. To me, "throw up" is synonymous for "vomit." Doing a quick search yielded a definition from the Merriam-Webster and Cambridge dictionaries online. M-W: "2. Give up, quit." Cambridge: "UK informal - If you throw up your job, you choose to leave it or stop doing it."
However, like I said before, I've never heard it used that way. For resigning from work, the most common expressions I've heard are "left" and "quit."