The name of Jibril (also know as "Gabriel" in English) and Mika'il are indeed written in Quran. But the names of Azrael and Israfil are not. How do we know actual names of these angels? Did we Muslims imported angel names from other religions?
There is no mention tying that name to the angel of death in either Quran or saheeh Hadiths. The name Azrael comes from Jewish traditions incorporated later by some scholars as part of what is known as "Isra'eliat".
He is mentioned in the Quran as Mikal ميكال.
The spelling Mika'il ميكائيل comes from a few disputed Hadith (see below) and Jewish traditions.
This name has only been mentioned in a few Hadiths such as this one: