No, at least not directly in any way that it can be understood directly from the text. What can be said most accurately is that the Qur'an does not conflict with DNA theory and therefore is in harmony with it.

However, some people read this idea into the story of Imam 'Ali (A) marrying the mother of al-'Abbas (A). Some people also read this into a narration about examining a woman's family before you marry her.

However there is no evidence that either of these things relate to DNA theory as we know it, and there is a tendency for some Muslims to try to find current ideas in texts when they aren't obviously there.

The idea that traits are inherited has been known to people since ancient times, especially since people have done farming and animal breeding and know well that the offspring resemble their parents in many ways, while at the same time, some traits skip a generation or are not always inherited.

Still, there are some Muslims who will argue that the Qur'an does discuss DNA, so you can read their arguments and the Qur'an and come to your own conclusion.