As of today, none of the major messengers offer post-quantum security for messages. Although it is confirmed that Signal has it on their roadmap and they’re working on it. There are some smaller messengers and projects working on this (e.g. Vox messenger) - use at your own risk and don’t trust, verify. For now, this section will only focus on the major messengers. We highly recommend you to use Signal for the highest level of security.
From the Time magazine's interview with Signal's president Meredith Whittaker
Meredith Whittaker on Signal and post-quantum security (December 4, 2022)
“We are investing in research to update the Signal protocol for post-quantum. It’s long term, slow burn research work that isn’t always publicized, but that we continue to do. But post quantum robustness for the protocol and the other encryption methods we use is definitely something we’re working on. The hope and the plan is to roll that out before it’s necessary, so there’s no gaps. I’m not a cryptographer, and I don’t spend all my time in the literature here. But I think the answer of ‘when’ is very much up in the air, based on what I do understand. There is a tech hype cycle playing fast and loose with terms like quantum computing, even as it’s very true that quantum is progressing, and we do need to be responsible and make the required investment to ensure we’re prepared.”