If you build your Java app using any implementation of Java 13 that passes the quality tests from Oracle (TCK) and/or the quality tests from Eclipse Adoptium (formerly AdoptOpenJDK) (AQA), then your Java app will run on any other implementation of Java 13 also passing those tests.
You certainly can develop your app on the Zulu JVM from Azul Systems while deploying to your customer’s Oracle JVM from Oracle, as they have both passed quality tests to verify they comply with the Java platform specifications.
I’m addressing the technical angle. You and your customer must also abide by the licensing terms of any Java implementations you obtain.
You also have the option of bundling a JVM within your app, as long as you abide by licensing terms. Research tools such as jlink.
Be aware that Java 13 is end-of-life as far as I know. Consider using the current version 14, or the current LTS version 11.
You have a choice of several vendors offering builds and installers based on source code from the OpenJDK project. Here is a flowchart I made to help guide you in choosing one.

And here is a list of considerations in choosing a vendor.

You said:
because I heard that oracleJDK from now has to be payed
Do not make important business and technical decisions based on what you "hear". Read the legal terms to which you are agreeing by the download and usage of software such as a Java implementation.