The time complexity of finding the diameter of a graph
I assume you mean the diameter of $G$ which is the longest shortest path found in $G$.

Finding the diameter can be done by finding all pair shortest paths first and determining the maximum length found. Floyd-Warshall algorithm does this in $\Theta(|V|^3)$ time. Johnson's algorithm can be implemented to achieve $\cal{O}(|V|^2\log |V| + |V|\cdot|E|)$ time.

A smaller worst-case runtime bound seems hard to achieve as there are $\cal{O}(|V|^2)$ distances to consider and calculating those distance in sublinear (amortised) time each is going to be tough; see here for a related bound. Note this paper which uses a different approach and obtains a (slightly) faster algorithm.