Most markets do not pay on acceptance. And why would they? Most editors want to make sure the story you write is something they will actually use especially if they don't know you. Newspapers are quicker than magazines for the most part, but you probably already knew that. Writing is really not something to get rich quick at. What's the hurry?
The magazines I've sold stories to have paid within 7 days after I signed the contract. I think that's quick pay ~ but the acceptances take time.
I've been paid to write ad copy, guest editorials, advice columns, and local interest essays within a week of print, but those were jobs I got based on personal connections and business contacts.
Some content writing gigs can be pretty profitable if you get the right ones and get assigned to a special project. I've written for Demand Studios for a couple years. It comes and it goes, but if you keep your scores up and have some special expertise, you can make out okay. I don't know if they are taking on new writers right now, but I make $25 for short, 500-word articles. Not great pay, but the format is really simple. It's pretty easy to make $100 a day there if you're on a special project, and they pay twice a week -- Tuesdays and Fridays. I've written for them on and off for a couple years, and my pay has never been late.
If by markets you mean magazines, it's hard to get paid fast. Even if you get paid on acceptance, you still have to query, wait for them to respond, sign a contract, write the story, go through edits and then get paid. That can take months. If what you're looking for is quick money, then you have to focus on web content, which doesn't pay too well but can be profitable if you're a quick writer.