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.
For SHA256 there is currently no known algorithm that would give quantum computers an edge above classical CPUs, let alone ASICs.
Grover's algorithm? Not that it will break hash functions completely, but it will hugely reduce their security, so bigger hashsizes would be needed.
it is worth knowing that the problem with hashes is always hash collisions not breaking the hashes. they are still irreversible. even SHA1 that became obsolete is still irreversible but it is possible to find collisions in it.
as for changing, i believe it is best to change the algorithm instead of just increasing the size. for example using SHA3-256 instead of using a bigger size of SHA512