应该可以的。

His excuse why he forgot her birthday was that he had lost his diary.

No one believes his explanation why he made such a stupid mistake. 

以上两个句子是我自己写的，没有权威出处。

理由是excuse和explanation都隐含reason的意思。

excuse的意义之一为：a reason, either true or invented, that you give to explain or defend your behaviour.

explanation的意义之一为： a reason given for something.

但以上两个句子也可能不被接受。因为当先行词为reason时，关系副词why的意义为for this/that reason,  而现在先行词为excuse或explanation时，关系副词why的意义就成了for this/that excuse/explanation. 用其作从句的原因状语则逻辑不通。

因此，为了安全起见，还是应该避免用excuse和explanation作why定语从句的先行词。可以改为其它结构。如上面的两句可以改为：

His excuse for forgetting her birthday was that he had lost his diary.定语从句改为for+动名词结构。

No one believes his explanation  for why he made such a stupid mistake. 定语从句改为for+名词性从句结构。

赞同Mr.Tong老师解答观点！

why引导定语从句先行词只能是reason不能是其他词。

(1)在reason作先行词的定语从句中，若从句表原因，一般用why引导定语从句，why可以用for which替换。why还可省去。
The reason why (for which) he has resigned is poor health.

他辞职的理由是健康状况不佳。
(2)当reason作主语带有定语从句修饰时，其后的表语从句通常用that不用because。
The reason why he didn't come in time was that he didn't feel well.
他没有来的原因是因为他不舒服。
(3)当reason用作定语从句的宾语时，引导词用that/which,而且that/which可省去。
The reason (that/which) he gave for being late was that they were held up in a traffic jam.
他给出的迟到原因是由于他们被交通阻塞给耽误了。