Can I become an intern

You need to apply for a residence permit in Denmark if you want to do an internship in the green sector (agriculture, forestry, and gardening). 
You must be at least 18 years old, but not older than 30 at the time of application, and you must pass a language test at a minimum A2 level on the Common European Framework of Reference for Languages (CEFR) scale and provide documentation of your ongoing education.

The internship must have a professional connection to the education you have started in your home country or in another country where you have resided legally. This means that the internship in Denmark should complement your ongoing education.
SIRI will assess your educational documents in relation to your internship tasks. You apply for a residence permit through SIRI, where you can also find more information about the conditions and rights.

If you are a citizen of an EU/EEA country or Switzerland, you do not need to apply for a residence permit but instead apply for an EU residence document. Link to the Agency for International Recruitment and Integration, SIRI.