You can combine forward DP (from source) and backward DP (to destination) to count paths that pass through a specific node. Compute forward[u] = paths from source to , and backward[u] = paths from to destination.
The number of paths through node is forward[u] * backward[u]. Sum over all nodes to count total paths, or find the node that maximizes forward[u] * backward[u] to find the most popular intermediate node.