Looking for Du'aa made in Masjidul-Haraam

The only dua mentioned in the Hadeeth with regards to the Night of Power (laylatul Qadr) is narrated by Aishah (radhiallaahu anha):

Allaahumma innaka 'Afuwwun Kareemun tuhibbul 'afwa fa'fu 'annee

اَللَّهُمَّ اِنَّكَ عَفُوٌّ ، تُحِبُّ الْعَفْوَ فَاعْفُ عَنِّي

O Allah You are The One Who pardons greatly, and loves to pardon, so pardon me.
[at Tirmidhi]

How ever, as you are looking for Du'a e Qunoot, this is what we recite

(TRANSLATION)
Bismillahir Rahmanir Rahim
I seek help from you, ask forgiveness from you, and believe in you and rely in you and praise you for all the good things and are grateful to you and we part and breack off with all those who disobedient to you. O Allah, you alone do we worship and pray exclusively to you and bow before you alone and we hasten nagerly towards you and fear your severe punishement and hope for your mercy, for your severe punishement is surely to be meted out to the disbelievers.
--------------------------------
ALLAHHUMMA INNA NASTAINOKA WANASTAGHFIRUKA WA NO MINO BIKA
WA NATAWAKKALO ALAIKA WA NUSNEE ALAIKAL KHAIR WA NASHKOROKA WALA NAKFOROKA WA NAKHLAO WA NATROKO MAIYN YAF JOROK. ALLAHHUMMA IYYA KANA BUDO WALAKA NUSALLEE WA NASJUDO, WA ILAIKA NAS AA WA NAHFIDO WA NARJOO RAHMATAKA WA NAKHSHA AZABAKA, INNA AZABAKA BIL KUFFARE MULHIQ

The Dua performed in the Haramain vary and there are many many duas they recite.

This is the most common Dua they start with.

اَللَّهُمَّ اهْدِنَا فِيْمَنْ هَدَيْتَ

Allaahummahdinaa feeman hadait (Aameen),


وَعَافِنَا فِيْمَنْ عَافَيْتَ

Wa 'aafinaa feeman 'aafait (Aameen),


وَتَوَلَّنَا فِيمَنْ تَوَلَّيْتَ

Watawallanaa feeman tawallait (Aameen),


وَبَارِكْ لَنَا فِيْمَا أَعْطَيْتَ

Wabaarik lanalla feemaa a'atait (Aameen),


وَقِنَا شَرَّ مَا قَضَيْتَ

Waqinaa sharra maa qadait (Aameen),


فَإِنَّكَ تَقْضِيْ بِالْحَقِّ وَلَا يُقْضَى عَلَيْكَ

Fa-innaka taqdie walaa yuqdaa 'alaik (Haqqaa),


وَإِنَّهُ لَا يَذِّلُ مَنْ وَاْلَيْتَ، وَلَا يَعِزُّ مَنْ عَادَيْتَ

Wa-innahu laa yadhillu man waalait, walaa ya'izzu man 'aadait (Haqqaa),


تَبَارَكْتَ رَبَّنَا وَتَعَالَيْتَ

Tabaarakta rabbanaa wata'aalait (Yaa Allaah),


لَكَ الْحَمْدُ عَلَى مَا قَضَيْتَ

Lakal-hamdu 'alaa maa qadait (al-Hamdulillaah),


نَسْتَغْفِرُكَ اللَّهُمَّ وَنَتُوْبُ إِلَيْكَ

Nastaghfiruk-allaahumma wanatoobu ilaik (Astaghfirullaah).

Oh Allaah guide us along with those whom you have guided,

And pardon us along with those who you have pardoned,

Be an ally to us along with those whom you have allied with,

And bless us Oh Allaah with that which you have bestowed upon us,

And save us from the evil of what you have decreed,

For indeed you decree and none can influence you,

And he is not humiliated whom you have befriended, nor is he honoured who is your enemy,

Blessed are you, O Lord and Exalted,

To you is all the Praise on what you have decreed,

We seek your forgiveness Oh Allaah and turn in repentance to you.

Here is another version