Matthews occupies third place in Glasgow's all-time scoring charts - only George Horne and DTH van der Merwe can better the hooker's tally of 51 tries for the club.
Matthews crossed for a hat-trick in a 40-12 home victory over Zebre Parma in May 2022; the feat made him the first hooker ever to score three times in one game for the club, as well as becoming the first Glasgow player to score a hat-trick since Nikola Matawalu away to the Ospreys in 2018.
He repeated the feat against the same opposition a year later, scoring another hat-trick in the same fixture as the Warriors earned a 50-8 win over Zebre Parma in March 2023.
Matthews then went on to create club history against the Dragons at Scotstoun, crossing for five tries in a single match - a feat never before achieved by a Glasgow Warrior.
He finished the 2023/24 BKT URC season as top try-scorer, breaking his own club record by scoring 14 tries to top the standings in an historic title-winning campaign for the Warriors.
The former Boroughmuir man became the club's first-ever try-scorer at Hampden Park in December 2024, touching down for a brace in a 33-14 win over Edinburgh.
Matthews reached a century of appearances in a 49-0 win over the Dragons in October 2025, marking the milestone with a try off the bench.
US Montauban announced the move today and confirmed Matthews has a deal that will take him through to 2029.