In Agents of S.H.I.E.L.D. Coulson and his crew are shown to be very wary of Agent Triplett when they discover that Agent Garrett was a Hydra agent.
Why does nobody suspect Agent Ward, especially when everybody knows that they share a long history?
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The S.H.I.E.L.D team had been together a long time, all the way through the first season. During that time, Ward had risked his life for different members of the team on multiple occasions.
This included throwing himself out of a plane, and killing the Hydra agents when him and Skye were hidden in the closet together.
At this point, I'm guessing everyone thought that he was beyond reproach.
Sure, Ward had a long history with Garrett when they were partners, but Coulson had known Garrett even longer. It would have been entirely feasible for him to have not known Garrett's true intentions.
I think it was harder for them to believe that Triplett was innocent because he had been his most recent partner. It would make sense that as Garrett's plan came closer to fruition, he would surround himself with allies rather than enemies.