Quran does allow travelling into the future

BUT not the past.

A lot of people fantasise about what regrettable errors they would change in the past with a time machine.
There is a major problem with this idea.

If you do travel into the past and change your mistake there is no way in the future you would ever remember the mistake because it never was a mistake to begin with.

To give an example imagine a car trip where turning left leads you to a traumatising future with problems and turning right was a safe passage.

If you do travel to the past and give yourself the idea to go right instead of left (as you did before), instantly the future you would forget the mistake of the past and at the future time machine the future you would be contemplating another mistake of the past to change, instead of this left/right decision. That error would never have existed so it wouldn't be a choice at the machine.

Let's say the time machine couldn't make you change drastic decisions but only allowed you to make a minute change in the past, something as little as not having a certain thought or even moving a hand. This would cause the entire future to change entirely. You could not exist as you now because everything in our past shapes us. Something even as simple as changing one thought in the past would change everything. 

Allah has established a governing law and that is time moves forward.

Travelling into the past is impossible.
"Be and it is" means everything has been set before the creation was even created. It could never be "Be and it was"

However, the Quran clearly mentions travelling into the future is possible.
Albert Einstein, in his theory of special relativity, determined that the laws of physics are the same for all non-accelerating observers, and he showed that the speed of light within a vacuum is the same no matter the speed at which an observer travels

Here is Einsteins Special relativity in the Quran (All movement is relative to other objects.)

1#
2:259 Or like the one who passed by a township, and it (had) overturned on its roofs. He said, "How (will) bring to life this (town) Allah after its death?" 
Then he was made to die (by) Allah (for) a hundred year(s), then He raised him. He said, "How long (have) you remained?" He said, 
"I remained (for) a day or a part (of) a day." 
He said, "Nay, you (have) remained one hundred year(s). Then look at your food and your drink, (they did) not change with time, and look at your donkey, and We will make you a sign for the people. And look at the bones how We raise them, then We cover them (with) flesh." 
Then when became clear to him, he said, "I know that Allah (is) on every thing All-Powerful."

Allah is the truth, Math is the numerically visualised language of the truth and science is the physical entity study of the truth. Allah will never go against the truth in order to prove his all powerful.

If time travel into the past was possible, Allah would have granted the request from the man into going back and seeing the township coming back from the dead, but instead, Allah makes him travel hundred years into the future and visually see his donkey being put back together. 

2#
27:38-42 He said, "O chiefs! Which of you will bring me her throne before that they come to me (in) submission?"
Said a strong one of the jinn, "I will bring it to you before [that] you rise from your place. And indeed, I am for it surely, strong, trustworthy."
Said one who, with him (was) knowledge of the Scripture, "I will bring it to you before [that] returns to you your glance." Then when he saw it placed before him, he said, "This (is) from (the) Favor (of) my Lord, to test me whether I am or I am ungrateful. And whoever (is) grateful then only he is grateful for his own soul. And whoever (is) ungrateful, then indeed, my Lord (is) Self-sufficient, Noble."
He said, "Disguise for her her throne; we will see whether she will be guided or will be of those who are not guided."
So when she came, it was said, "Is like this your throne?" She said, "It is like it." "And we were given the knowledge before her and we have been Muslims."

In the Suliaman narration, we find with the proper knowledge an item can be made to travel distances at the speed of light.

3#
3:49 And (make him) a Messenger to (the) Children (of) Israel, "Indeed, I [surely] [I] (have) come (to) you with a sign from your Lord that 
I design for you from [the] clay like the form (of) the bird, then I breath into it and it becomes a bird by (the) permission (of) Allah. 
And I cure the blind, and the leper, and I give life (to) the dead by (the) permission (of) Allah. And I inform you of what you eat and what you store in your houses. Indeed, in that (is) surely a sign for you, if you are believers.

Here Isa has the power to make a bird from the shape of clay with his breathe (with the Holy Spirit). This could potentially mean the clay in Isa's hands didn't just become a bird but it could easily be "evolved" into a bird within his hands. The clay could be like the man and the donkey in 2:259 merely speeding up time. 

4#
3:59 Indeed, (the) likeness (of) Isa near Allah (is) like (the) likeness (of) Adam. He created him from dust then He said to him, "Be," and he was.

A common misconception is Adam and Isa were created from nothing. This ayat emphasises that they were not created from nothing but from dust (or a combination of worldly produces materials/compounds). 

This could mean as Allah can (and does) speed up time, he could have easily evolved Adam and Isa without the normal requirements of reproduction. 

So to summarise There is no going back, we can only move forward. Stop living in the past and make sure you take right steps so you don't have to dwell on it in the future