if you have trusted certificates set up, and you can log into browser and your Chrome or Firefox is not complaining about the SSL, then your certificates should be good.
A thing to check is your NEO4J_URI. If you consult our drivers manual, there was a recent change which introduced new schemes like neo4j+s, neo4j+ssc, and so on. Those schemes can affect the way that certificates are processed, so it's something to look into.
Another thing to double check is that Neo4j separates the SSL policy of bolt and https. In the article, you'll notice settings like dbms.ssl.policy.bolt distinct from dbms.ssl.policy.https. This makes it possible to get https right, but still not have trusted certs in place for bolt, if things aren't set up correctly. You have to have the certificates set correctly for bolt in order for the client connection to work.
Note that if you use a URI scheme like neo4j+s://localhost in combination with a driver flag (trust all certificates) the URI scheme you're using still wins.