First off... don't be fooled by "Bitcoin stocks". Most are only Bitcoin-related or at the very most are companies that pool investments to buy and manage Bitcoin for you. I understant there could be a market for people who don't understand or have the time or desire to learn how to buy or store their own Bitcoin... but I cannot comprehend how the stock premiums are more attractive.
So technically... you can't quote Bitcoin.
Is Bitcoin a stock? - No
Can it be quoted on a "stock" exchange? As it isn't a stock, you can guess the answer.
However, there was a tradition of quoting all sorts of things unofficially on the old London Stock Exchange, so maybe there are some private listings. Don't forget that there is a Bitcoin futures market, and that is intended to be used by Bitcoin price speculators ( and the pumpers and dumpers ).
Yeah, Sweden's been offering Bitcoin derivatives for years. I recall trying to do a bit of research about this some time last year but I never got back responses from the people I contacted so didn't pursue it. Yes, that's a disclaimer for the info I'm sharing here.
https://xbtprovider.com/#product is a sample of such a product... you get all the same information. The cert is actually a Bitcoin tracker, this was the first product offered and still trades on Nasdaq Stockholm. Unlike the US futures, they hold coins equal to value of certs issued.