I don’t think there is anything by default, the logging is pure python, but python has a flexible framework for logging and a quick chat with GPT suggests it is possible to create a custom postgres logger for python. This would be outside of the meltano framework though. I suspect you have already checked out the logging section of the docs, but here’s the link just in case:
https://docs.meltano.com/guide/logging/#logging
This would actually be pretty interesting to try…