Do you power your broadband off when you aren't using it? It's possible that the V6 boxes are temporarily waking up to do some housekeeping, but aren't able to complete it because they have no broadband connection.
The other reason this can happen is if there is a stray infra-red noise in the room that the box thinks is someone pressing remote control buttons (but it would be unusual for this to happen in two rooms in the same house). If this is the cause, it is easily resolved. As long as the remote control is paired with the V6 box, the IR receiver on the front of the box isn't used at all, so you can block it up with a bit of black tape or cardboard, or even stick the V6 in a closed cupboard. The receiver is in the black shiny panel just to the left of the VM "infinity" logo.
Remote control pairing instructions are here
https://www.virginmedia.com/help/vir...ring-tv-remote