The correct option is D Adam says that he has a headache. He can't go.
Direct speech is a sentence in which the actual words of the speaker are quoted.
Indirect speech is a sentence in where we report what someone said, without quoting his exact words.
If the reporting verb is in the Present Tense, the tense of the verb in the reported speech remains unchanged.
Here, option D is correct. Reporting verb 'says' is in present tense, so tense in reported speech doesn't change. 'I' changes to 'he' as Adam is talking about himself.
In option A pronoun doesn't change. So, it is incorrect.
In option B tense changes. So, it is incorrect.
In option C can changes to could, which is wrong in this case. So, it is incorrect.