Thanks for the replies guys! The good news is this client is a server admin so they know tech stuff AND they keep image backups of their stuff, so I am waiting for a backup before their problem started occurring so I can compare files. Basically the issue is when a page is viewed for the first time, a database connection error is shown (it's a WordPress error, not a PHP error); however, when you refresh the page, it loads just fine. If you click off the page to another one, same thing, then you hit refresh and it's okay. I'm thinking it has something to do with caching plugins or something with a cache. It just comes at an interesting time because I made an addition to their site to add a shortcode for him, so he just wants to make sure it wasn't something I did, but I don't see how that would affect database connections.
In any case, the copy of the files I have from when the site was experiencing issues has confirmed malware on it (or at least remnants of it) that looks linked to the cache plugin(s), so once I am able to compare the backup files with the problematic files, I can confirm it wasn't me. They are running a very ancient version of WordPress as well which doesn't help anything (and is probably the reason the site has issues).
Once again, thanks for the replies!