Unlikely as it would not be much of a proof. When you apply for a passport there is a thorough review of your identity documents but the proof of travel is only seen at the point of application or order to determine that you are eligible for the service and collect the higher fee.
Nobody can tell you for sure what the agent will do that day for your application. From experience though, they simply ask to see the tickets or an itinerary. It's usually a printout and so they certainly know it would be easy to fake, just as it would be easy to have someone answer the phone telling them you will travel some time soon.
The issue with car travel is that you do not usually have proof or a fixed time. It would be best if you make some booking that shows your intent to be in the US by a certain date. Fully refundable hotel fares are quite common. You can also book another form of travel, mostly on buses from Canada and the US, but possibly a train ticket if there is such service close to where you are. Some may not be refundable, or not fully, but this would provide some level of proof.