The Holy Quran gives three kinds of arguments on this point.
Firstly, it refers us to the physical world which shows great order and arrangement, works according to laws, and where everything has a set purpose in the whole scheme of things. There is also immense beauty in nature which attracts man's heart. Science is discovering more and more of these characteristics of the world all the time. So behind this highly purposeful and beautiful working of nature must be One, single Intelligence of great beauty and attraction.
Secondly, the Quran tells us about the close and deep connection between God and the inner nature of every person. There is an in-built desire in each and every person to search for something higher than oneself, and when in difficulties a person instinctively wants to call upon a Hidden Power to help him.
Thirdly, and most convincingly, Prophets and men of God appeared in every nation in the world and showed their people the reality of God through their teachings and works. Just as most of us learn about science not by making all the discoveries ourselves, but by accepting the work and evidence of those who devote themselves to the study of science, similarly the proof of the existence of God is provided by the lives of the great luminaries whom God sent all over the world for this very purpose.
17:10
9:25
26:38
2:40
2:54
5:39
2:32
30:40