Terry Bywater top-scored for GB with 20 points, but they could not win a fourth successive title as Turkey won 76-69.
GB's men had trailed by 12 at half-time and fought back to within three in the final quarter, but Turkey - who won the pool match between the teams - held on to win the gold medal.
Earlier in the day, GB's women won bronze with a 68-37 win over France.
They won every quarter in the bronze-medal match, having beaten the same opponents by four points in the pool stages, with co-captain Helen Freeman top-scoring with 27 points.
GB women's coach Miles Thompson said: "The game we played against France was the game we were searching for this tournament.
"We separated early and we stayed separated through consistent defence."
The women's final was won by the Netherlands, who beat Germany 56-46.
GB men's co-captain Terry Bywater said: "We'd won it three times - we wanted a fourth. I think the first twenty minutes we were nervous, we wanted it so bad and it showed.
"We showed that we are one of the best teams in Europe. At the end of the day it's about keeping our heads held high - we made the final."
Great Britain's men lost to Turkey in the final of the European Wheelchair Basketball Championships in Tenerife.