The 23-year-old right-back moved to the Stadium of Light in a £2million deal after winning four Scottish titles with the Hoops.
Matthew's last appearance for Wales came in a 2-1 World Cup qualifying defeat in Macedonia couple of years ago and he is desperate to turn the head of Dragons boss Chris Coleman by performing at Premier League level.
Matthews told Sky Sports: "The international thing was a major part of me moving.
"The league in Scotland isn't great and I think the Welsh manager saw that and hopefully this move will help me improve my chances of getting into the Welsh team because they are doing really well at the minute. But I just needed a new challenge."