Not out of the box. There are a couple ways to get the message log, but in both cases, it'll require some heavy parsing of the logs to get the messages you want to capture and write it out to your SQL DB.
You can either enable a setting in the Alteryx System Settings under Engine > General and add a path in the Logging Directory field. Anytime the server runs a workflow, the message log will be written to that location and named with the timestamp (EPOCH formatted) for when the job started.
The other option would be pulling it directly from Alteryx Servers MongoDB, which can be a bit tedious if you're not familiar with the collection schema.
The Alteryx Server Usage Report, attached, is a good starting point and can also help understand how to query data and parse the message log and customize it to fit your needs. Note that this utility has been deprecated after 2021.4 but I think you should still be able to use to grab the message logs.
