Neo4j user trying to implement it with GraphQl and use it in a Flutter app

I think you should change your URL uri: 'http://127.0.0.1:7474/graphql/' to your local ip-address it is gonna be something like (http://192.168.1.109: 7474/graphql). Check your network preferences for the ip-adress and change your 127.0.0.1 with what you have.