Vincenti's stunning 30-yard strike helped the Shots to a 3-2 second-round victory against Fleetwood on December 1, the day of Cole's death.
The 27-year-old is hoping his effort can beat the five other nominees to the award, as a tribute to the passing of his close friend.
He told Get Hampshire: "It would be fitting for the award to be dedicated to someone of his calibre.
"It's not been an easy week. I was meant to meet up with him this coming Monday. We're good friends so finding out the news was just heartbreaking.
"He'll be sorely missed by a lot of people."