The 29-year-old midfielder scooped both the fans' and players' accolades ahead of Danny Kedwell in second and Paulo Gazzaniga in third.
"I'm over the moon," Jackman told the club website.
"I had a couple of tough years beforehand and it's nice in my third year to get these awards, it means a lot to me.
"It's nice, but I'd give it up for promotion, any player would because that is what you are judged on ultimately."
Gazzaniga was named young player of the year after an impressive debut season with the Gills.