I think they are believed to be, but the opposite is not true. There are a lot of exponential or factorial problems that have no efficient QC algorithm, and are not beleieved to be likely to have one.
Also, at least the most famous problem which has an efficient QC algorithm, integer factorization, has no proven lower complexity bound on classical computers. That is, while the best algorithm we know for it is exponential (or, technically, sub-exponential), it is not yet proven that no polynomial classical algorithm exists. The problem isn't even NP-complete, so it's not like finding an efficient algorithm would prove P=NP.
Also, at least the most famous problem which has an efficient QC algorithm, integer factorization, has no proven lower complexity bound on classical computers. That is, while the best algorithm we know for it is exponential (or, technically, sub-exponential), it is not yet proven that no polynomial classical algorithm exists. The problem isn't even NP-complete, so it's not like finding an efficient algorithm would prove P=NP.