Complexity of counting all connected subgraphs
Welsh states that the problem #P-complete even in the most restricted case (counting the number of connected subgraphs of a planar bipartite graph). See the bottom of page 305 in Welsh, Dominic (1997), "Approximate Counting", Surveys in Combinatorics, Bailey, R. A., ed., Cambridge University Press, pp. 287–324.

In context, though, I wonder whether he really means connected spanning subgraphs. And that leads me to wonder which version of the problem you want: connected spanning subgraphs, connected subgraphs that need not be spanning, or connected induced subgraphs?