As others have alluded, when lettered, you want to represent your carrier appropriately, so you wouldn't want to use the vehicle in a manner that could damage their reputation. It's your personal vehicle, so you can use it as such, within reason. Like, you can't be home and OOS and haul freight or conduct other for-hire business with the lettering still on the truck. If tracking when you are OOS is a problem, then unplug the QC. However, do keep in mind that when you lease on with a carrier, you are literally transferring the right to possession and/or use of goods or equipment (your truck) for a term in return for consideration. Unplugging the QC for a lengthy period of time could be a breach of the contract, since they have the right to know where the QC is at all times.
Of course, if you're going to take your truck somewhere and do something
really illegal, then you're better off risking the breach by unplugging it. And turning off your phone and wrapping it in tin foil.