Q. Is it safe to assume that there are no reports in the Qur'an performed by the Prophet himself?
Nothing good comes from assumptions, there is one miracle mention in the Qur'an the is linked to prophet Muhammad (PBUH). This miracle is the splitting of the moon, Qur'an 54:1 says:
The Hour (of Judgment) is nigh, and the moon is cleft asunder. - [Qur'an 54:1]
Q. Is the same true for the Hadith?
That the Meccan people requested Allah's Messenger (ﷺ) to show them a
miracle, and so he showed them the splitting of the moon. - Sahih al-Bukhari 3637 Book 61, Hadith 141
So that should answer both of your questions and to be honest a quick Google search would have given you a lot of information.