Well here's a discovery, the checkered fabric Burke 116 in TOS "Court Martial" doesn't have the triangles.
But the pleather/vinyl upholstered Burke 116's do.
I didn't notice if the checkered fabric Burke 116 at STST had them but the one in STC's "Embracing the Winds" does.
I always thought the triangles looked really sharp on the chairs and am still planning to include them on my checkered fabric Burke 116 build even if they aren't strictly canon. It's possible TOS may have added them later or maybe the checkered fabric back and seat cushion were put onto a different Burke 116 if they should happen to show up in another episode. Not certain if the back of this chair style can be seen in other episodes though.