Hi there, great question! Articles can be really tricky, but there is also another interesting part to your example sentence: the main verb.
I am guessing that you are talking about one specific holiday that will start in the near future, is that right? You could say "I don't want to learn German during the holidays.", but using 'learn' sounds like you think you will finish/complete learning German during the holidays. Compare these examples:
"I want to learn how to juggle during the summer holidays."
"I don't want to learn how to play football this summer, I want to learn how to play tennis instead."
-----> These are things you will finish learning.
"I want to study a little bit of English everyday during the holidays."
"I want to improve my English speaking this summer."
"I don't want to work on grammar all summer, that would be so boring."
----> These are all things you are already learning, and which you can improve but can't really finish learning.
So a more natural version of your sentence would be:
"I don't want to have to study German during the holidays."
"I don't want to spend my holidays practicing German."
"I don't want to work on my German during the holidays."
Hope this is useful and interesting :)