Is there any verse in the Coran that shows that Hijab is obligatory?
There are these two ayah:
And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed. -- Qur'an 24:31
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful. -- Qur'an 33:59
Also relevant is Qur'an 24:60: "...women of post-menstrual age who have no desire for marriage - there is no blame upon them for putting aside their outer garments [but] not displaying adornment...".
The most relevant ahadith I'm aware of are:
(the wife of the Prophet) 'Umar bin Al-Khattab used to say to Allah's Messenger "Let your wives be veiled" But he did not do so. The wives of the Prophet used to go out to answer the call of nature at night only at Al-Manasi.' Once Sauda, the daughter of Zam'a went out and she was a tall woman. 'Umar bin Al-Khattab saw her while he was in a gathering, and said, "I have recognized you, O Sauda!" He ('Umar) said so as he was anxious for some Divine orders regarding the veil (the veiling of women.) So Allah revealed the Verse of veiling. -- Narrated 'Aisha (sunnah.com)
This gives some background as to the motivation behind observing veil.
May Allah bestow His Mercy on the early emigrant women. When Allah revealed: "... and to draw their veils all over their Juyubihinna (i.e., their bodies, faces, necks and bosoms)..." (V.24:31) they tore their Murat (woolen dresses or waist-binding clothes or aprons etc.) and covered their heads and faces with those torn Muruts. -- Narrated 'Aishah (sunnah.com)
This hadith indicates how important it is considered to observe. (Other ahadith are listed on Islam Q&A.)
Is hijab considered fard?
As you might expect (given the diversity of hijab seen in the Muslim world), there is diversity of opinion of what constitutes hijab. As to whether or not hijab is fard (obligatory), I'm yet to see a scholar disagree that it's fard:
The first commandment applies to women when they go out. Then they are to use an outer garment which should cover their whole body. -- Malik Ghulam Farid (ed), The Holy Qur'an, Arabic Text with English Translation & Short Commentary, 2003 (pdf, within the discussion on pages 717-719; the author even includes covering the face as part of the commandment)
Almighty Allah ordered the women to let the two ends of their headgear extend onto their bosoms so that they conceal their ears, the neck, and the upper part of the bosom also. -- Al Islam, which cites Ar-Razi, at-Tafsiru 'l-Kabir, vol. 23, p. 179
This is a command from Allah to the believing women, and jealousy on His part over the wives of His believing servants. It is also to distinguish the believing women from the women of the Jahiliyyah and the deeds of the pagan women. -- QTafsir.com
Sheikh Yusuf Al Qaradawi... also states that all of the scholars throughout history from various groups such as the Sufis, the Dhahiris (the Literalists), the fuqahaa’ (jurists), and the scholars of hadith unanimously agree that it is fard for the believing woman to cover her hair. -- Virtual Mosque
And I personally observe hijab because of (a) what it says in the Qur'an, and (b) I'm unaware of any reasonable dissenting opinion.