number of pair of vertices with at least 2 distinct paths between them in a undirected graph
I'm assuming that you're looking for the number of pairs of vertices having two disjoint simple paths connected them. A pair of vertices has two disjoint simple paths connecting them if they are in the same biconnected component. You can partition the graph into biconnected components in linear time. If the biconnected components have sizes $N_i$, then the number of pairs is $\sum_i \binom{N_i}{2}$.