This monastery, "Girls monastery" (I don't know if this refers to nuns, or to tome other feature) dates back to 1349-1390 and is also known as the Panaghia or Theoskepastos monastery. It houses some fresco's that I show, but you will see they are badly ruined.
I take this information from a good site by the Trabzon municipality, I edited it somewhat: Built on the outskirts of Trabzon-Boztepe this is the only remaining monastery in the city centre. It was first built by Alexious III. Additions were made in the 18th and 19th centuries. Inside are the church built for the archbishop of the monastery, Konstantinos, and his tomb, as well as guest rooms and student rooms. The buildings surrounding the courtyard used to be the service buildings next to the main church, a monks cell, student rooms, the bell tower in the yard and another church belonging to a later period and the guest rooms on the outermost. The monastery complex was built on two terraces and is fortified by high walls. Actively used until the early 20th century and later abandoned.
To the left the rock church, straight on up the stairs another church, under which there is a cell where people were locked in for 40 days of penitence and fasting, the entrance to the monastery is to the right of that.