Generally speaking, no. If you are a Danish citizen who has always lived here, or if you have taken Danish citizenship after living here for many years, you are ineligible for subsidised Danish language education at a language school. The Danish language education is for foreigners, and Danish citizens are, by definition, not foreigners, no matter how inadequate their Danish language skills are. VUC offers courses for Danish citizens who need to work on their language skills.
If Danish is your native language, and you need help with writing skills, Studieskolen's Business department has courses in Danish for Danes.
Find more information.
There are exceptions for Danish citizens who have lived most of their life abroad and did not have the opportunity to attend Danish schools when they were young. (For example, you have a Danish parent but grew up in another country and have only just come to Denmark.) Danish citizens from Greenland and the Faroe Islands are also exceptions to this rule.