The story that you recall is supposed to have taken place in Haidamaque, Hungary, at some point around 1730. It comes to us at third or fourth hand, and the original account is almost absent researchable details; it seems there is reason to doubt the accuracy of those that do appear.

All in all, the story is probably best considered a doubtful anecdote that is practically unresearchable, but I'll let you have the details we do know.

The original source of the story that you saw summarised by the History Channel is Augustin Calmet, a French monk who put together a noted casebook which remains of considerable importance in vampire studies: his Dissertations sur les apparitions des anges, des démons et des esprits, et sur les revenants et vampires de Hongrie, de Bohême, de Moravie et de Silésie was published in 1746, and an English edition appeared in 1850 as The Phantom World.

Calmet was a scholar, and attempted to examine the reports of vampires, witches and angels that he published with some scepticism, but he interpreted rather than investigated the cases that he'd collected. The full account of what went on in Haidamaque is fairly short, so I'll excerpt it for you here; the translation (by Henry Christmas) is a pretty poor one, but otherwise the detail matches pretty well with your recollections:

About fifteen years ago, a soldier who was billeted at the house of a Haidamaque peasant, on the frontiers of Hungary, as he was one day sitting at table near his host, the master of the house saw a person he did not know come in and sit down to table also with them.

The master of the house was strangely frightened at this, as were the rest of the company. The soldier knew not what to think of it, being ignorant of the matter in question. But the master of the house being dead the very next day, the soldier inquired what it meant.

They told him that it was the body of the father of his host, who had been dead and buried for ten years, which had thus come to sit down next to him, and had announced and caused his death.

The soldier informed the regiment of it in the first place, and the regiment gave notice of it to the general officers, who commissioned the Count de Cabreras, captain of the regiment of Alandetti infantry, to make information concerning this circumstance. Having gone to the place, with some other officers, a surgeon and an auditor, they heard the depositions of all the people belonging to the house, who attested unanimously that the ghost was the father of the master of the house, and that all the soldier had said and reported was the exact truth, which was confirmed by all the inhabitants of the village.

In consequence of this, the corpse of this spectre was exhumed, and found to be like that of a man who has just expired, and his blood like that of a living man. The Count de Cabreras had his head cut off, and caused him to be laid again in his tomb.

He also took information concerning other similar ghosts; amongst others of a man dead more than thirty years, who had come back three times to his house at meal-time.

The first time he had sucked the blood from the neck of his own brother, the second time from one of his sons, and the third from one of the servants in the house; and all three died of it instantly and on the spot.

Upon this deposition the commissary had this man taken out of his grave, and finding that, like the first, his blood was in a fluid state, like that of a living person, he ordered them to run a large nail into his temple, and then to lay him again in the grave.

He caused a third to be burnt, who had been buried more than sixteen years and had sucked the blood and caused the death of two of his sons.

The commissary having made his report to the general officers, was deputed to the court of the emperor, who commanded that some officers, both war and justice, some physicians and surgeons, and some learned men, should be sent to examine the causes of these extraordinary events. The person who related these particulars to us had heard them from the Count de Cabreras, at Fribourg in Brigau, in 1730.

So Calmet sets down an account that he claims to have had from an anonymous person who heard them in turn from an officer who actually investigated the case, and had met the witnesses, but had not personally been present when the events that you are interested in supposedly occurred. The only details that we can actually check are the names of the – very vaguely located – village and the investigating officer, and that of the officer's regiment. I've been able to find no sign that a village called "Haidamaque" actually exists in Hungary (and the name scarcely sounds Hungarian, either), but I do note that the Romanian "haidamac" translates as "thief" or "scoundrel" and can be found in some late 18th/early 19th century sources as an identifier for groups of brigands in the Carpathians, which is not very encouraging. The latter two details do not check out either; Leo Ruickbie, who is the author of the only scholarly account I'm aware of that actually mentions the case, notes that "there was no Cabreras of the Alandetti, though an identification with Juan de Cabrera i Perellós of the Spanish noble family of Cabrera, serving in Anton Diego, Marquis D'Alcaudete's regiment, seems likely."

Ruickbie concludes his discussion of the story by noting that "the involvement of military personnel, including legal and medical officers, gives the story apparent authority, but at the present time it is unsubstantiated," but it is worth pointing out, in closing, that the description of the disinterral of the corpse of the suspected vampire actually is a very close match for accounts given in several of the much better authenticated cases studied by Paul Barber in his influential Vampires, Burial and Death (1990). As Barber points out, it is actually not that uncommon for bodies in certain states of decomposition to have signs of blood around the mouth (and to appear to continue to grow fingernails, another detail often commented on in such cases).

Sources

Paul Barber, Vampires, Burial and Death: Folklore and Reality (1990)

Leo Ruickbie, "Evidence for the undead: the role of medical investigation in the 18th century vampire epidemic," in Barbara Brodman & James E. Doan, The Universal Vampire: Origins and Evolution of a Legend (2013)