It was not the first time Bond has been suspicious of someone, nor the last. He certainly was so with the driver from Governor's House in Dr. No, for example.
I think what gets lost in that From Russia with Love scene is the immediate class distinction. Grant is not a svelte, dignified British gentleman but a hulking brute, His walk, his bearing, his expression all betray a quality of lower class lack of refinement barely concealed in a suit. Contrast Grant with the real Captain Nash, who is more fitting of the type. Grant is more like an American bodybuilder who kicks sand in the face of the 99 pound weakling. Perhaps it is not as obvious because Connery himself is more brutish than, say, a David Niven or Cary Grant type. But Bond is disdainful toward Nash all along, and Bond's instincts are true.