Is 6 months the minimum time in which I can be an intern?

I believe 6 months is the minimum contract term with TaLK (6 months being pretty flexible -- most private English academies and other public school programs only offer 1+ year contracts). That said, you could try and touch base with the program's administration to see if they'd make an exception. You could also try the Private Academy / "Hagwon" route and see if they offer summer term contracts.

recently looked this up for my cousin who was interested, and I think that the terms have changed from when I was there. For some reason the website isn't working for me at the moment. The terms are at fixed times: for example you may begin either in March or August (if those are still accurate, check the website to be sure), so most likely just teaching for the summer won't work for this particular program. Another option you can look into, if you currently have any connections in Korea, is see if there is a family who would like to hire you for the summer. That's what my family wanted me to do because they thought I would make more money, but I personally felt more comfortable going with this program.