You'll need to get an alias on the relationships/edges if you want to include them in your json output, one approach to this is to construct a subgraph (copied from the linked post) -
MATCH (n)
WHERE n.user_id='0000001'
CALL apoc.path.subgraphAll(n, {maxLevel:1}) YIELD nodes, relationships
WITH [node in nodes | node {.*, label:labels(node)[0]}] as nodes,
[rel in relationships | rel {.*, fromNode:{label:labels(startNode(rel))[0], key:startNode(rel).key}, toNode:{label:labels(endNode(rel))[0], key:endNode(rel).key}}] as rels
WITH {nodes:nodes, relationships:rels} as json
RETURN apoc.convert.toJson(json)
Original post & explanation -