That's a bit broad for a forum question, but here are a few options for your problem:
1) If the API is available openly you could call it directly from within Mendix, see https://docs.mendix.com/howto/integration/consume-a-simple-web-service
2) The alternative is using indeed java, there's a few constraints that Mendix puts on the Java actions it creates, but aside from that you can create your own classes and import your own libraries, see https://docs.mendix.com/howto/logic-business-rules/extending-your-application-with-custom-java
Hope this helps
Judging from your Postman call, it looks like you are trying to make a REST Get call. This is possible in Mendix and you need to use the Mendix REST microflow actions. Check out this documentation for examples on how to call an API using a similar construction to your Postman:
You can always use the Nexmo App that's available in the Mendix App Store. This will simplify most the work for you.
We're using it at Brighton & Hove City Council for our text messaging, and it's very simple.
You can get a Java action to call a microflow, so you can do most the setup in Mendix before calling it from Java.