I'm sure the other large banks have some kind of online instant payment service as well, so I've updated the thread title.

You could always do interbank wire transfers, which can get expensive in the US, though they are common in much of the rest of the world. There are also bill payment services through banks, where you can pay regular bills like utilities and such.

Online instant payments like quick pay seem like they are set up more for the kinds of uses that PayPal addresses, enabling two ordinary people who are not operating a business to make a transaction via e-mail.