Forster, 23, had a year remaining on his contract with Hibernian, who won the Scottish Championship in 2016-17.
However, the former Plymouth loanee has left Easter Road in search of regular first-team football.
"It's great to have a player with as much experience as him at the age of only 23," manager Gary Johnson told the club website.
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Defender Jordon Forster has joined English League Two side Cheltenham from Scottish Premiership club Hibernian.