Border guards could suspect that you have taken the car without permission. For that matter, so might police if you get stopped for any reason. You could try to rely on your ability to persuade them that you have your girlfriend's permission. You could also try to rely on your ability to reach your girlfriend by telephone. In the vast majority of cases, this strategy will succeed.
It would be safer, however, and probably faster, to have written permission. It would be safer still if that written permission were in the form of a notarized letter. Many banks offer notary services free of charge; some even do so regardless of whether you are a customer of the bank.
Here's what the US CBP has to say on the matter:
If the car or vehicle is loaned or being brought into or taken out the United States on behalf of a friend or relative, the driver might want to have a notarized copy of a permission letter addressed to the "Officer In Charge of Customs and Border Protection" to drive the car across the border.
(Source: https://help.cbp.gov/app/answers/detail/a_id/248/~/taking-%2F-bringing-a-car-%2F-vehicle-into-or-out-of-the-u.s.-that-belongs-to-a)
I didn't find anything from CBSA about this; the only information I could find concerned Canadian residents. But it shouldn't hurt to address the letter to both the CBSA and CBP, or to all law enforcement generally. I would most likely address it "to whom it may concern."
