There has been some literature discussing this and a migration strategy:
- Giechaskiel, I., Cremers, C., Rasmussen, K.B. (2016). On Bitcoin Security in the Presence of Broken Cryptographic Primitives
- Stewart, I., Ilie, D., Zamyatin, A., Werner, S., Torshizi, M.F., Knottenbelt, W.J. (2018). Committing to Quantum Resistance: A Slow Defence for Bitcoin against a Fast Quantum Computing Attack
- Ciulei, A.T., Crețu, M.C., Simion, E. (2022). Preparation for Post-Quantum era: a survey about blockchain schemes from a post-quantum perspective
The 2nd referenced paper describes a commit-reveal scheme that would avoid having your funds stolen when you want to migrate them to some new, quantum-resistant address.
The new addresses would pretty much look the same: a string of characters, with maybe few bits of difference in starting characters to encode the use of some new scheme. If collision resistance is required they'd also have to be a little longer (384 bits). Quantum preimage resistance is already achieved with SegWit 256-bit addresses (even though the underlying key is vulnerable).
So addresses wouldn't be affected much. However, transactions would get much bigger, since the input script would then have to include a bigger public key and signature. For example, SPHINICS uses 1KB keys and 41KB signatures so in that case the address would have to be about 7 times longer and signature about 645 times longer! Good news is that with SegWit it wouldn't count against the "hard" blocksize limit, and that data could be later pruned.