Yes the practice of wearing Hijab is required by the Quran. We find the injunction in the following verses from the Quran:
“Say to the believing men that they should cast down their glances and guard their private parts…say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what must ordinarily appear thereof” (Qur’an 24:30-31)
“O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) over their bodies (when out doors). That is most convenient that they should be known and not molested.”(Quran 33:59)
As the Quranic verses were revealed to the Prophet Mohammad (SAWS), he would recite them to the Muslims and the believers would immediately accept them and follow them. Thus it became a customary practice among Muslim women to use the hijab and cover themselves modestly.