It was most likely a simple preserve.
Unripe walnuts are commonly eaten in Eastern Europe, and preserve is one of the widespread preparations. (Maybe even the only widespread one besides liqueur - they are mostly eaten raw). I don't know about Georgia, but this certainly exists in other countries. It is a somewhat exotic preserve, compared to more common ones like strawberries, but something ordinary people know of (at least people of a jam-making generation).
For the preserve, the walnuts are cooked in sugar syrup and then sterilized, similar to other fruit like figs. Since the whole walnuts are used, the texture is indeed crunchy - the protoshell is left within the nut during the preparation, and while it isn't toothbreaking-hard yet, it does give a crunchy bite.
The way you describe it, it seems that it was served pure, to be eaten with a spoon. This is also typical for Eastern Europe and other cuisines with Ottoman influences. It is not the only way to eat jam and preserves, but it may be surprising for people from cultures where less sweet desserts are preferred.
For some background, you can read the Wikipedia page for this type of whole-fruit preserve (it is not specialized about the walnut preserve). Apparently, when cooked in the Caucasian region, the local word is "murabba".
