The phrase “just happened” has two meanings. It can mean that the thing happened quite recently. It can also mean that it happened, but that there was no cause or reason for it to happen. In other words, when we say it “just happened” we mean that it was just a random event, with no purpose and no person purposefully causing it to happen.
I assume you are asking the second question. In other words, you are asking if the big bang was caused. The answer is of course it was caused. This is explained in what is known as the kaalam cosmological argument. It goes like this:
Premise: Everything in the universe which begins to exist was caused.
Premise: The universe began to exist.
Conclusion: The universe was caused.
In other words, it did not “just happen.” I know of literally not a single exception to the first premise. I have not even ever heard of anyone proposing an exception to premise #1 that makes any sense at all. Premise #2 is accepted by all cosmologists. Both the observable facts (red shift, cosmic microwave background radiation, distribution of created elements) and the second law of thermodynamics requires that the universe had a beginning. Almost certainly (but we cannot exactly “prove” it), it was begun in an event we call the big bang.
So, the universe was caused. It did not just happen. In general, nothing literally just happens. Those who say that the universe just happened had better either explain more carefully what they mean by just happened, or they had better give us an example of anything that, literally just happens without a cause. In the universe we live in things do not just happen.
No, the universe did not just happen.