Sweden-born Bergqvist moved to England at the age of 10 and came through the youth ranks at Reading and QPR.
The 20-year-old centre-half arrives at St James Park as a free agent following his departure from Aldershot.
City chief executive Julian Tagg said: "We're delighted to have had the opportunity to bring in a young prospect of Doug's calibre.
"Not only does he have exciting potential for the future but he has already played league football and knows what the game is about at a competitive level.
"He is someone we've looked at the possibility of signing before and we're very pleased we have had the opportunity to add him to our ranks."