Hi @joe_gerner,
Thanks for your question! Unfortunately writing to SLD files isn't supported yet, but there is an idea here for it if you would like to upvote it and comment with your use case.
I took a look at the accepted formats in QGIS and noticed that it does accept MapInfo layers if you wanted to look into that. In this QGIS documentation if you scroll down a bit, there's a blog post titled "New Tool: MapInfo to QGIS style converter" that you could explore.
There's been some speculation in the community about SLD being an XML file and that it might be possible to manipulate the XML if you are very familiar with the SLD structure. Most of the previous cases though in wanting to write to SLD are going between QGIS and Geoserver.
Lastly in my search, this answer from GIS Stack Exchange suggests that it's not easy to store MapInfo styles within QGIS, and likely requires a plugin. Hope that helps!
- Jovita