The proofs that we have for Allah's existence are nothing but various claims mentioned repeatedly in the Quran. These claims come under three categories:
1- Direct announcements
1.1 Allah's prime announcement describing himself as eternal, absolute, unique , having no offspring , nor is he born from anything and there is none equal to him (Chapter 112)
1.2 Further general announcements in different verses where Allah claims that he exists and he possess many superior features highlighted in his ninety nine names mentioned in the Quran.
2- Allah's claim to be the creator and the prime controller of everything we see around us. (people, animals, plant, life, death, skies, stars, moon, sun, wind, cloud, rain, thunder, day and night). Allah specifically claims that he is the one that makes birds airborne capable of flying and make ships to get afloat and enable them to sail.
3- His challenges:
No one will be able to duplicate his Quran
No one but Allah is capable to bring a dead human or animal back to life
No one can help people when they are in need except him.
Undeniable acknowledgement from people saying;"it is Allah who created us" if were asked who created them.