First of all, your friend needs to correct his understanding of Islam. You can not rely on Quran only and say that you do not need the Hadith. Quran has made it abundantly clear though multiple Verses that obedience to the Prophet is of utmost necessity. No one can enter Paradise by disobeying the Prophet Salallahu Alaihe Wa Sallam. The whole purpose of sending the Prophet with the Book was so people could understand the Book from the Prophet and practice it accordingly.
Quran does not address everything and Quran certainly does not address the details of almost all the things with the exception of a few. You can take Prayer for example. The command to establish Prayer has been mentioned repeatedly in the Quran but the method of offering the Prayer and establishing it is not mentioned in even a single place throughout the Quran. For that we need the guidance of the Holy Prophet Sallallahu Alaihe Wa Sallam.
In conclusion, without the Hadith, practice of religion would be impossible.
In answer to your question, there are clear statements from the Holy Prophet Sallallahu Alaihe Wa Sallam that prohibit the practice of tattoo.
This prohibition is an enough evidence for a person to give up on this practice. In addition to that, the prohibition of this practice may be understood from Verse 119 of chapter 4.
Therefore tattooing is forbidden in Islam.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America