What should I do regarding a relative of mine who makes melodic errors (laḥn) when reciting Surah Al-Fatihah and says that he has not learned Tajweed?

If a relative makes a clear error in reciting Al-Fatihah that alters its meaning, it is obligatory to explain the danger of this to him, and that it invalidates the prayer, along with gently and kindly teaching him the correct recitation. If he turns away after the explanation, then you have fulfilled your duty.

However, if his error is in omitting recommended aspects of recitation (such as elongations), the matter is easier and does not affect the validity of the prayer. He should still be encouraged to learn so that his recitation becomes more complete and perfect.