I assume you are not currently a member of an A-kasse, so it is important you apply at latest 2 weeks after graduating to be able to get your graduate-rights in the A-kasse.
Regardless of your current job is paid or unpaid, you can't receive full dagpenge while having the job. You must be fully available for the labor market, and you are not if you have the internship.
Off course you have the option to get supplementary benefits (supplerende dagpenge), but its only possible for maximum 30 weeks, and all the hours spend in your job (paid or not paid) will be deducted in the amount you get in supplerende dagpenge.
If you will avoid the risk of getting a 3-week quarantine because of self-inflicted unemployment, you must stop your work (internship) BEFORE you apply for A-kasse membership. If you have a notice period, you must not be in the notice period when applying for dagpenge. You must actually have stopped working.
Anders Weber, www.a-kasser.dk