The youngster has been rewarded with a new deal after impressing this season.
The youngster signed a new four-and-a-half-year deal on Thursday that means he is tied down at Goodison Park until at least 2020.
It has been a rapid rise for the former MK Dons man, who has starred in the side at left-back in the absence of Leighton Baines this season.
It has been quite the journey since he made his debut against West Ham United at the end of last season, and he says Martinez wanted to reward him for his progress so far with fresh terms.
“The gaffer told me he wanted to reward me and I was more than happy to commit my future,” he told evertonfc.com.
“As soon as I got here I knew this was the place that I wanted to be for a long time. With the aspirations that the Club has for the future, I want to be a part of what’s happening here.
He also says that the potential within Everton’s squad at present was one of the key factors in him agreeing a new deal.
Galloway is one of a number of youngsters currently starring in the Blues first team, with the likes of John Stones, Ross Barkley, Gerard Deulofeu and Romelu Lukaku all impressing this season.
Their futures are also looking bright, given their respective development at present, and Galloway expects them to be a real threat in the years to come.
“With the other young talent that we have got, if we all stick together then we can be a real threat in the future,” he added.
“There is a lot of excitement within the squad over what we can achieve. We look a threat in every game, we haven’t lost too often, but even then we want to improve and get even better. We all have that hunger to succeed.”