This seems not to be the case for me here and now. When you register your free trial on odoo.com. If I go to my account, I get option to choose the registered company or to create a new one. The registered free company has an arrow down, where one of the the options is "download backup". If you click there ,it says something like, that my odoo.com user is not admin, so it seems you cannot do this operation by your own. Please correct me, if I am wrong. Also adding  /database/manager to url  just throws a 404 on odoo.com quick-start-instances.  (also compare this screenshot)

I also read this thread dealing about same topic, but I get forbidden on the suggested url:

https://www.odoo.com/forum/help-1/question/where-is-the-control-center-how-to-get-a-backup-of-my-database-solved-68631

I think that basically requires 2 steps.

1. backup your online database (logout from your Odoo > manage databases > backup
2. restore the database in you on-premise installation > manage database > restore

I haven't done that ever, so I am not sure what problems might occur. I guess it is important that your on premise installation has the same modules installed as your online version.

There will be other ways to do that with PostgreSQL tools as well.