Hi @ninabreznik
While I can’t recommend any particular Payroll package, I can advise that in terms of connections to QuickFile, The Payroll Site does have this ability.
The Payroll Site will allow you to run payroll, carry out RTI for HMRC, and then export your figures into QuickFile. However, this software isn’t free, and as far as I’m aware, doesn’t have an API, although you may wish to contact them directly to confirm.
There’s more information on this setup, here:
Of course, this is a public forum, so other users may share their experiences of other packages below