I'm developing a program that use MongoDB database but I'm asking if it is possible to use an internal database for test like H2 when doing SQL through Java?
You can try to use Testcontainers
Testcontainers is a Java library that supports JUnit tests, providing lightweight, throwaway instances of common databases.
Or look at https://github.com/flapdoodle-oss/de.flapdoodle.embed.mongo
Update:
It seems that there is already an answer to your question: Embedded MongoDB when running integration tests