I think the bridge crew were aware something was off with their Captain almost from the get-go because of his uncharacteristic behaviour. Lester/Kirk was certainly unnecessarily rude to Spock almost right off and in front of the entire bridge crew.
And in fairness it's not impossible that Kirk could address an officer or crewman by their position or rank rather than their name. If he wasn't immediately familiar with a new face it's an easy out when addressing them. He also might use their rank or position depending on circumstance.
To Sulu: "Helmsman, I want a right angle course."
To Lt. Leslie (in "This Side Of Paradise"): "This is mutiny, Mister."