It should be “Anna, Jenny isn’t studying for her exams.” The comma usage is correct here.
A comma should be placed after the person being addressed, which is Anna. In this sentence, the direct address is at the start of the sentence. This is correct.
A single quotation mark (“) does not work here because the entire sentence seems to be spoken aloud, and the placement does not make sense. This is incorrect.
A colon (:) does not make sense here because a colon is used to separate 2 independent clauses if the second (after the semicolon) explains or illustrates the first (before the semicolon). This is also incorrect.