Fortunately, the maximum block size for legacy nodes is 1 MB, and for Segwit nodes is 4 MB. If someone wants bigger blocks than 4 MB, then by running the current version, you simply won't see their data. And in general, I think no matter if block size will be bigger or smaller than today, quantum signatures should not be processed by existing nodes, because they don't know, how to handle it (and different quantum proposals may have different needs). Unless you know, how to re-write any quantum signature as a bunch of OP_CHECKSIG operations, along with other Script opcodes, then it could be visible by non-quantum nodes.

Also, I guess if OP_CHECKSIG will be really broken, then it could be possible to activate quantum signatures on top of OP_CHECKSIG directly. Because then, if quantum signatures use some 256-bit numbers internally, then OP_CHECKSIG can be used as their 256-bit calculator, and then, any public keys could be used anywhere, because going from public to private key can be just part of the process. Because even in Shor's algorithm, going from public to private key doesn't have zero cost: even if it is solvable, then it still require N quantum operations. Which means, that by creating dependencies between public keys and signatures, it can be made much harder, than just "<pubkey> OP_CHECKSIG".