The Blue Square Premier side boss was handed the chairman's car as a gift for leading his team to an 8-1 victory over Barrow on Saturday following their 7-0 win last March.
"I didn't really take him at face value when we was talking but to be fair to the man he's a top guy and when he did it, it was unbelievable," Cox told BBC Radio Five Live.
Radford said: "It's my old Aston Martin so it's not brand new. It's a couple of years old. You have these things and you've got to give people targets and things like that, so he fancied the car and I thought I'm about ready for a change and he did the job."
The car, which had personalised MT51 COX registration plates put on it, is thought to be worth £80,000.