Since then, the full-back has been an important part of the Premier League club's squad.
However, his playing time decreased significantly last season, with Veltman scoring one goal in 24 league appearances, nine of which came from the start.
He had also been entering the final months of his contract with the Seagulls.
Brighton have now confirmed that Veltman is leaving the club. Brighton chairman Tony Bloom was clear about the impact the defender has had during his time at the Amex Stadium.
"Joel has been an amazing player during his six years with us," Bloom told the club's official website. "We knew we made a really good signing when we bought him, and he has delivered both on and off the pitch just as we hoped.
"He has been an important player during our positive development during his time here, and played a number of matches during the season when we qualified for Europe for the first time."
It remains unclear where the 34-year-old defender will play his football next season.
"We are very grateful to Joel and wish him all the best for the next step," Bloom added.
Veltman made 191 appearances for Brighton across all competitions since joining from Ajax, having arrived on the south coast for a reported £900,000 fee. He also served as a Brighton & Hove Albion Foundation ambassador, a role he took on in 2022, and was a regular supporter of the club's community initiatives throughout his time in Sussex.
Brighton finished eighth in the Premier League last season, a result that means the club will play in the Conference League next term.
Veltman's departure brings to an end a six-year association with the club that included regular first-team football and a role in one of the most significant periods in Brighton's recent history. His experience and versatility across the backline made him a reliable option for successive Brighton managers, and he leaves with the club's gratitude for his contribution both on the pitch and within the dressing room.
A move away from the Amex Stadium would give Veltman the chance to continue his career elsewhere, with his next destination still to be determined as he enters a new chapter following his release.