Short answer
No, the evidence points the other way. First, science: the universe is expanding. In 2003 three physicists showed that a universe which has, on average, been expanding cannot have been doing so for ever. Its past has an edge. Second, reason: if the past had no beginning, an endless number of days would have passed before today. An endless series can never be completed, so today could never have arrived. Al-Ghazali made this argument nine centuries ago. Third, even an eternal universe would not explain itself. A universe made of parts and ruled by laws still depends on something for its existence. Every road leads to a being that exists by its own nature, whom the Qur'an calls the First and the Last.
Full explanation
The evidence of a beginning
In 1929 Edwin Hubble reported that the farther away a galaxy is, the faster it is moving away. In 1965 Arno Penzias and Robert Wilson picked up a faint, steady signal from every direction in the sky, the afterglow of a hot early universe. Then in 2003 Arvind Borde, Alan Guth and Alexander Vilenkin proved that a universe which has, on average, been expanding cannot have been expanding for ever. Its history must have a boundary in the past. The physicist Arthur Eddington had said in 1931 that, as a philosopher, he found a beginning repugnant. The evidence pointed to one all the same.
Reports that the farther away a galaxy is, the faster it is moving away from us: the observation that became the evidence for an expanding universe.
Reports a faint, steady radio signal at 4080 megacycles coming equally from every direction in the sky, which they could not trace to their equipment. It was recognised as the cosmic microwave background, the afterglow of the hot early universe, and earned them the Nobel Prize in Physics in 1978.
Shows that a universe which has, on average, been expanding cannot have been expanding for ever: its expanding history must have a boundary in the past.
Eddington writes that, speaking as a philosopher, he finds the idea of a beginning of the present order of nature repugnant.
An endless past could never reach today
Suppose the past had no beginning. Then before today an endless number of days would already have gone by, one after another. But a series made by adding one day to the next can never be completed if it has no first member, any more than someone can finish counting down from infinity. If such a series could never be completed, today could never arrive. Yet today has arrived. So the series of past days had a beginning. Al-Ghazali argued this in The Incoherence of the Philosophers, and the argument is known today as the kalam argument, after the Muslim theologians who made it.
In its first discussion, argues against the philosophers' view that the world has no beginning, including the argument that an infinite series of past events is impossible.
Traces the argument that the universe began and so has a cause to the Muslim theologians of the kalam tradition, above all al-Kindi and al-Ghazali, and defends it in modern form.
Infinity in mathematics and in the world
A common reply is that mathematics works with infinite sets, so an infinite past is possible. But an idea that can be defined is not the same as a series of events that has actually happened. David Hilbert, the mathematician who defended the theory of infinite sets, concluded that the infinite is found nowhere in reality. Its place, he said, is as an idea that mathematics uses. Mathematics lets us describe an endless line of numbers. It does not let a line of real days be walked to its end.
The mathematician who defended Cantor's theory of infinite sets concludes that the infinite is found nowhere in reality. It does not exist in nature, he says, and it gives no sound basis for reasoning about what is real. Its role is that of an idea, which mathematics uses.
Even an eternal universe would need a reason
Suppose, for the sake of argument, that the universe had no beginning. It would still be made of parts, governed by laws, and one way rather than another. Ibn Sina and Leibniz argued that whatever does not exist by its own nature needs a reason for its existence outside itself, however long it has existed. A book copied from an earlier copy for ever still needs an explanation for why it says what it says. So the question of why there is a universe at all does not go away with time.
Divides all that exists into what is only possible in itself (mumkin al-wujud), which needs a cause, and what is necessary in itself (wajib al-wujud), which needs none. Argues that a chain of possible things, however long, needs a necessary being outside it, and that this being is one and has no parts.
Argues that the sufficient reason for the whole series of contingent things must lie outside that series, in a necessary substance, which Leibniz calls God.
Islam's answer: the First and the Last
The Qur'an calls Allah (﷾) the First and the Last, the Evident and the Immanent. He alone is without beginning. Everything else, the heavens and the earth among them, was brought into being by Him. The Qur'an also asks people to reflect on the creation of the heavens and the earth, and calls it a sign for people of understanding.
“He is the First and the Last, the Evident and the Immanent: and He has full knowledge of all things.”
“To Him is due the primal origin of the heavens and the earth: When He decreeth a matter, He saith to it: "Be," and it is.”
“Behold! in the creation of the heavens and the earth, and the alternation of night and day,- there are indeed Signs for men of understanding,- Men who celebrate the praises of Allah, standing, sitting, and lying down on their sides, and contemplate the (wonders of) creation in the heavens and the earth, (With the thought): "Our Lord! not for naught Hast Thou created (all) this! Glory to Thee! Give us salvation from the penalty of the Fire.”
The replies answered
Borde, Guth and Vilenkin, Physical Review Letters (2003)
Shows that a universe which has, on average, been expanding cannot have been expanding for ever: its expanding history must have a boundary in the past.
Cited for
The Big Bang is only the beginning of our expanding phase. Something may have existed before it, so the theorem does not show a beginning of everything.
The response
That is right about the theorem: it shows that the expanding history has an edge, not what lies beyond it. But whatever lies beyond it is either another part of a physical series, which raises the same question of an endless past, or something that is not physical and depends on nothing. Pushing the beginning back does not remove the need for one.
Where it lands — Moving the beginning further back still leaves a series that cannot be endless, or a cause that is not physical.
Eddington, Nature (1931)
Eddington writes that, speaking as a philosopher, he finds the idea of a beginning of the present order of nature repugnant.
Cited for
Saying the universe began is just a religious preference dressed up as science.
The response
The resistance has often run the other way. Eddington, a leading astronomer, said he found a beginning repugnant, yet the observations of Hubble, Penzias and Wilson, and the theorem of Borde, Guth and Vilenkin, all point to one. The argument here rests on those findings and on reasoning anyone can check.
Where it lands — The beginning was accepted because of the evidence, not because of religion.
Perspectives
The universe may have no beginning
Some physicists and philosophers hold that the universe, or something before it, may have always existed. They point to proposals in which time has no first edge, or to cycles of universes.
Hartle and Hawking proposed a model of the universe in which time has no first edge. So the universe need not have a beginning.
Rebuttal
It is a proposal, not an established result. Even on it the universe is finite in the past, and a universe made of parts and laws still needs a reason to exist.
Mathematics deals with infinite sets, so an infinite past is not impossible.
Rebuttal
An infinite set is defined, not built step by step. The past was built one day after another, and such a series can never be completed. Hilbert himself held that the infinite is not found in reality.
Our universe could be one in an endless series of expanding and contracting universes.
Rebuttal
The Borde-Guth-Vilenkin result applies to any history that has, on average, been expanding, so it limits these models too. And an endless series of universes is still a series that depends on something.
The Islamic perspective
The universe began, and even if it had not, it would still depend on something. Only Allah (﷾) is without beginning: the First and the Last.
A universe that has, on average, been expanding has a boundary in the past.
A series of past events added one after another cannot be endless, or the present could not be reached.
Allah (﷾) is the First and the Last, without beginning, and He brought the heavens and the earth into being.