The R package sna has a read.dot() function that returns an adjacency matrix. From there, you can write the json output you want.

Here's a way to do this in Python:

#!/usr/bin/env python

import sys
import pygraphviz
import json

def main():
    G = pygraphviz.AGraph()
    G.read("foo.dot")
    adj = { 'adjacency_matrix_rows' : [] }
    for n in G.nodes():
        row = { n : [G.has_edge(n, tn)*1 for tn in G.nodes()] }
        adj['adjacency_matrix_rows'].append(row)
    sys.stdout.write("%s\n" % (json.dumps(adj)))

if __name__ == "__main__":
    main()

Given an example foo.dot:

digraph {
                a -> b;
                b -> c;
                c -> d;
                d -> a;
                d -> c;
        }

The Python output looks like:

{"adjacency_matrix_rows": [{"a": [0, 1, 0, 0]}, {"b": [0, 0, 1, 0]}, {"c": [0, 0, 0, 1]}, {"d": [1, 0, 1, 0]}]}

This JSON-formatted string is a JSON object, which contains an array of JSON objects, each of which is a key-value pair indicating the node name and its array of binary adjacency values.

Per the classic adjacency matrix, a 1 means there is edge between the object key ("first node") and the dereference of the second key's index ("second node"), directed from the first node to the second. A 0 means no edge.

If you're on OS X, I wrote a post on how to install pygraphviz.

If you have some Python chops, you can use the Pygraphviz API to build a Python object of dicts and arrays, converting the nodes and edges of your graph into exactly the JSON-formatted output that you need.