SQL Server full database backup without a job
A backup command can be run from just about anywhere and scheduled using any kind of scheduling tool. So, while SQL Agent is the most common method, it's not the only one. To determine what's happening when this backup runs, I'd suggest creating an Extended Events session to capture the queries. You'll be able to see the login that is being used. You may, depending on the connection settings, also be able to see the application making the call. At the very least, you can see the machine within your network that the call is coming from.
Because Extended Events can return a lot of data, I'd suggest turning it on only for the time that the calls are being made based on the log information you have. To search through the data, you can use the Live Data Explorer window within SSMS. I have a blog post on getting started with that tool that might help.