In Hadith to the prophet Muhammad (PBUH) he identified arrogance as "denying the truth and cut short people rights (deeds)". Non-believers denied God and his messengers for a lot of reasons. See the verses below:
1- Kinship: over history kings were competing with God for pride. Which is very stupid but arrogance is the first enemy of wisdom:
2:258 Hast thou not Turned thy vision to one who disputed with Abraham About his Lord, because Allah had granted him power? Abraham said: "My Lord is He Who Giveth life and death." He said: "I give life and death". Said Abraham: "But it is Allah that causeth the sun to rise from the east: Do thou then cause him to rise from the West." Thus was he confounded who (in arrogance) rejected faith. Nor doth Allah Give guidance to a people unjust. أَلَمْ تَرَ إِلَى الَّذِي حَآجَّ إِبْرَاهِيمَ فِي رِبِّهِ أَنْ آتَاهُ اللّهُ الْمُلْكَ إِذْ قَالَ إِبْرَاهِيمُ رَبِّيَ الَّذِي يُحْيِـي وَيُمِيتُ قَالَ أَنَا أُحْيِـي وَأُمِيتُ قَالَ إِبْرَاهِيمُ فَإِنَّ اللّهَ يَأْتِي بِالشَّمْسِ مِنَ الْمَشْرِقِ فَأْتِ بِهَا مِنَ الْمَغْرِبِ فَبُهِتَ الَّذِي كَفَرَ وَاللّهُ لاَ يَهْدِي الْقَوْمَ الظَّالِمِينَ
2- Commonly they see God signs in the universe, supranatural and scripture and they deny them:
7:36 But those who reject Our signs and treat them with arrogance,- they are companions of the Fire, to dwell therein (for ever). وَالَّذِينَ كَذَّبُواْ بِآيَاتِنَا وَاسْتَكْبَرُواْ عَنْهَا أُوْلَـَئِكَ أَصْحَابُ النَّارِ هُمْ فِيهَا خَالِدُونَ
3- Commonly they see the signs of God and they deny them though their hearts believe in them:
27:14 And they rejected those Signs in iniquity and arrogance, though their souls were convinced thereof: so see what was the end of those who acted corruptly!
وَجَحَدُوا بِهَا وَاسْتَيْقَنَتْهَا أَنفُسُهُمْ ظُلْمًا وَعُلُوًّا فَانظُرْ كَيْفَ كَانَ عَاقِبَةُ الْمُفْسِدِينَ
4- For some arrogance does not even let them read the Quran or examine God's signs:
31:7 When Our Signs are rehearsed to such a one, he turns away in arrogance, as if he heard them not, as if there were deafness in both his ears: announce to him a grievous Penalty.
وَإِذَا تُتْلَى عَلَيْهِ آيَاتُنَا وَلَّى مُسْتَكْبِرًا كَأَن لَّمْ يَسْمَعْهَا كَأَنَّ فِي أُذُنَيْهِ وَقْرًا فَبَشِّرْهُ بِعَذَابٍ أَلِيمٍ
5- Commonly arrogance allows them to continue to sin and do the evil things:
35:43 On account of their arrogance in the land and their plotting of Evil, but the plotting of Evil will hem in only the authors thereof. Now are they but looking for the way the ancients were dealt with? But no change wilt thou find in Allah's way (of dealing): no turning off wilt thou find in Allah's way (of dealing).
اسْتِكْبَارًا فِي الْأَرْضِ وَمَكْرَ السَّيِّئِ وَلَا يَحِيقُ الْمَكْرُ السَّيِّئُ إِلَّا بِأَهْلِهِ فَهَلْ يَنظُرُونَ إِلَّا سُنَّتَ الْأَوَّلِينَ فَلَن تَجِدَ لِسُنَّتِ اللَّهِ تَبْدِيلًا وَلَن تَجِدَ لِسُنَّتِ اللَّهِ تَحْوِيلًا
63:5 And when it is said to them, "Come, the Messenger of Allah will pray for your forgiveness", they turn aside their heads, and thou wouldst see them turning away their faces in arrogance.
وَإِذَا قِيلَ لَهُمْ تَعَالَوْا يَسْتَغْفِرْ لَكُمْ رَسُولُ اللَّهِ لَوَّوْا رُؤُوسَهُمْ وَرَأَيْتَهُمْ يَصُدُّونَ وَهُم مُّسْتَكْبِرُونَ
71:7 "And every time I (Noah) have called to them, that Thou mightest forgive them, they have (only) thrust their fingers into their ears, covered themselves up with their garments, grown obstinate, and given themselves up to arrogance. وَإِنِّي كُلَّمَا دَعَوْتُهُمْ لِتَغْفِرَ لَهُمْ جَعَلُوا أَصَابِعَهُمْ فِي آذَانِهِمْ وَاسْتَغْشَوْا ثِيَابَهُمْ وَأَصَرُّوا وَاسْتَكْبَرُوا اسْتِكْبَارًا
2:206 When it is said to him, "Fear Allah., He is led by arrogance to (more) crime. Enough for him is Hell;-An evil bed indeed (To lie on)! وَإِذَا قِيلَ لَهُ اتَّقِ اللّهَ أَخَذَتْهُ الْعِزَّةُ بِالإِثْمِ فَحَسْبُهُ جَهَنَّمُ وَلَبِئْسَ الْمِهَادُ