Now that we can see the data, we can see that payments is in fact an array (values uses []).

That means you need to call rootObj.getAsJsonArray("payments") which returns a JsonArray, and it is an Iterable<JsonElement>, which means your loop should be for(JsonElement pa : paymentsObject).

Remember, each value of the array can be any type of Json element (object, array, string, number, ...).

You know that they are JsonObject, so you can call getAsJsonObject() on them.

json = (json data)
JsonParser parser = new JsonParser();
JsonObject rootObj = parser.parse(json).getAsJsonObject();
JsonArray paymentsArray = rootObj.getAsJsonArray("payments");
for (JsonElement pa : paymentsArray) {
    JsonObject paymentObj = pa.getAsJsonObject();
    String     quoteid     = paymentObj.get("quoteid").getAsString();
    String     dateEntered = paymentObj.get("date_entered").getAsString();
    BigDecimal amount      = paymentObj.get("amount").getAsBigDecimal();
}

So you have the JsonArray object with your records, here's what you do to get your functional objects:

Type type = new TypeToken<List<DataMetrics>>() {}.getType();
List<DataMetrics> records = gson.fromJson(jsonArrayThatYouHave, type);

Then you iterate through you objects and filter the ones you need. In java 8 you can do this:

List<DataMetrics> result = records.stream().filter(record -> record.name.equals("Client::Sync")).collect(toList());

This approach is converting all objects and iterating after, if exactly this part of code is performance critical, you can still iterate through json and convert only necessary objects (but i doubt that this will be actually faster than described above).

Anyway this is more maintainable and understandable code.

UPDATE:

the same for java 7 will be:

List<DataMetrics> result = new LinkedList<>();

for(DataMetrics record : records){
   if(record.name.equals("Client::Sync")){
      result.add(record);
   }
}

You can simplify your code and retrieve the first_type array using the following code. Same code should pretty much work for the second_type array as well:

JsonArray types = map_json
    .getAsJsonObject("waypoints")
    .getAsJsonObject("ship")
    .getAsJsonArray("first_type";

for(final JsonElement type : types) {
    final JsonArray coords = type.getAsJsonArray():
}