A quarter gallery is an architectural feature of the stern

Stern

The stern is the rear or aft-most part of a ship or boat, technically defined as the area built up over the sternpost, extending upwards from the counter rail to the taffrail. The stern lies opposite of the bow, the foremost part of a ship. Originally, the term only referred to the aft port section...

of a sailing ship from around the 16th to the 19th century. Quarter galleries are a kind of balconies, typically placed on the sides of the sterncastle, the high, tower-like structure at the back of a ship that housed the officer's quarters. They functioned as a firing platform for the ship's marines and sharpshooters during boarding actions. The galleries also provided a structure that was ideally suited for attaching decoration and often bore carved wooden sculptures, particularly in the 17th century.