I dont believe any quantum computers that are out or coming out anytime soon can break SHA256 yet.
When something like that comes out, we should have some time to react before it starts breaking things.
Dont forget that a computer that can break SHA256 is a huge threat to the entire internet, not just bitcoin.
It is worth noting that the real worry regarding quantum computing and Bitcoin is not a possible breakage of SHA256 but of ECDSA, Bitcoin's transaction signing algorithm.
For SHA256 there is currently no known algorithm that would give quantum computers an edge above classical CPUs, let alone ASICs.
As far as ECDSA is concerned things could turn out bad though... eventually. It will likely still at least take a decade or two until quantum computers become a practical risk for ECDSA, if not longer. Either way it should be plenty of time to harden Bitcoin transactions against possible quantum computing based attacks.