Yes in blender triangles and ngons terminate edge loops, if you do a loop cut on your example and aim at an edge it will only add one vertex (creating two quads from the adjacent triangles.)
You can select the faces and use the “I” key (inset) which will create a loop.
Edit I deleted my second suggestion as the “I” key is the simple answer!
Have corrected Inset!
Since you have all of those edges selected, you can simply Subdivide. Or, if you really need a 3DS Max-ish “Connect”, there’s one in this addon:
@DNorman 'Tis Inset, not Insert.