The First Burying Ground of Hartford predates the Ancient Burying Ground.
There is no listing of graves for this cemetery, it was reclaimed and used by the city of Hartford. Many early settlers were buried there; those who died in Hartford at least before 1749, as that is the earliest grave listed at the Ancient Burying Ground.

The following information is now public domain, and provides an explanation.

Excerpt from "Hartford in the Olden Time" by Stuart Publishers, 1853

ITS FIRST BURYING-GROUND

... deposited in the manner we have described, in solemn keeping, at the corner of present Market Street and State House, Square.

Yes, Reader, there was the site of the first Burying-Ground of Hartford-an area close by the first Meeting-House, and running north from the Square towards the present City Hall, and west from present Market Street up, a little distance, the hill. It was then much more elevated than now-ten to twelve feet. It has been cut down since low enough to carry away all the dust both of the bodies and coffins of those who slept in its cold embrace. Its monuments, many of them, as they stood upon the site, were well remembered and frequently spoken of to his sons by the father of Messrs J. B. and C. Hosmer of this city, and also by the late Samuel Olcott. The latter said that many of the stones composing them were used in laying the foundations of some of the oldest buildings fronting on the Square, on it's north side.

But they are gone to our view—and the Settlers have transmitted to us none of their mummied bodies 'loaded with biography.' We have not even the name of a solitary one of those who were buried there. Their dust, distinguished or undistinguished in its day, is undistinguishable now. It is plain however from the customs and economy of the time, that those who first died in Hartford, did not retire to any resting-places that were made splendid with gravelled walks, and terraces and flowery banks, and costly trees and shrubs, like those of Mount Auburn, Greenwood, and our own North Cemetery-least of all to any magnificent villas like those of Pere la Chaise.

They were not laid up for eternity on shelves either of marble or of stone. No heaps of granite or of bronze, of lime or freestone, pedantic with inscriptions and chiselled with the exquisiteness of art-no pompous urns, or pyramids, or statues-no Winged Angels or Victories, wrought as if to make one think that " Hymen and Cupid, and not Death, walked through the Yard"—figured in Hartford's primitive Burying-Ground. But a few simple, upright stones, with plain inscriptions, and never any double Christian names-this would have been deemed a great innovation-possibly one or two somewhat massive slabs supported by five pillars on a foundation of stone, such as are seen now in old Grounds-these were the simple memorials erected over the graves of our first Settlers.

The turf was, green above these graves. It may also have been smooth, for the Town, judging from its subsequent practice in this respect, allowed horses and calves to feed upon its herbage. Yet there was a neat and 'sufficient' fence around, with pales and post heads 'handsomely shaped,' both for ornament, and to keep out the swine. Affection may have decorated some of the mounds, and doubtless did, with little shelters of shrubbery, with fern and woodbine and jessamine-there was plenty of these plants in the woods around. It may have strewn chaplets, and wreaths of flowers, over youth and innocence and beauty. It may have planted the cypress, the willow and the hemlock. But adornment was all simple, save that which nature perhaps furnished in some of the stately trees of the time, some gnarled oak, or tall whitewood, or majestic pines, through which the raven might caw in his plumage of crepe, and the breezes might sigh, or thunder storms roll their bass, or the clouds weep and the spheres play their sweetest harmony, above the sheeted dead.

— SCAEVA.