I found that almost nothing is written anywhere about the Church of the Virgin in Borjomi (where the cemetery is), although it is a cozy place on a hill with a beautiful view of the city. It's very quiet here!
The church itself was built in 1865, made of local stone with very beautiful carved doors. There is a small flowery garden, and from the panoramic platform there is a view of the Kura. Next to the church there is a small souvenir shop where you can buy cosmetics, which are made by the nuns themselves.
I don't know exactly what icons are there, but they say there is an icon of St. Nino, which is very revered in Georgia. And more relics, but I don't know whose.