The first case of PTSD was recorded around 1980, when the disorder was first defined. Before that, the concept did not exist. People knew about mental and physical problems related to war trauma, but this was not called PTSD and the specific group of symptoms we now call PTSD were not regarded as a coherent and singular disorder. /u/hillsonghoods explains the problem of definitions in his Monday Methods post on PTSD in past wars.

Since the term and the idea of PTSD did not exist before the Vietnam War, it is impossible for historians to tell whether warriors of past ages suffered from war trauma in the same way that modern soldiers do. We can't simply look at a description of a warrior's behaviour in a Medieval source and declare that what we're seeing is PTSD. We practically never have the level of detail required; our sources are not what we would consider medical diagnoses. Many cases from Antiquity are cited to "prove" that something we would call PTSD existed, but these identifications are always extremely informed by a desire to "find" PTSD, and really do not give anywhere near enough information for a diagnosis to be made. It is not historically or medically responsible to do so.

The posts I've linked outline the modern debate between scholars who argue that PTSD must be a universal response to war trauma and those who insist that war in the distant past was a meaningfully different experience that may not have triggered the same response. Neither side has been able to convince the other. More recent scholarship has tried to move past this debate, and to adopt a more historical perspective: did people in the past actually think there was something wrong with warriors when they came home from war? How did they interpret the behaviour of veterans? What specifically caused the changes in character and health that we hear about?

These sort of questions help us gain a better understanding of past responses to moral injury and the horrors of war. But they also mean that there can be only one answer to your question: we don't know who was the first to suffer what we now call PTSD, and we will never know. The first people to be diagnosed with PTSD were veterans of the Vietnam War, on whose behalf this specific disorder was defined.