So you just want to untriangulate? *keep the polys there but don't see the edge ( coz don't be mistaken, triangulatiung doesn't create any new poly, it just makes invisible edges visible )?
Then do as Vilkacis said. On how to do it, just to be sure:
1) click the mesh
2) push "2" ( the keyboard one, not the numpad one )
3) click on the edge you want to hide ( crtl+click to select multiple edges )
4) scroll down the modifier bar, on the very bottom, under surface properties ( I'll trust vilkacis on this one, but anyway, it's the very last thing on the bar, you can't miss it ), you'll find invisible. push it. voila, you won't see the edge anymore. You can also set an angle and push autoedge, if you're lazy and have a lot of edges to hide, but you'll probably still have a few edges to setup manually anyway.
edit: anyway, the way you explain it, my first way sounded right. welding the edges won't turn it into a triangle.
If you put back your original exemple pic, I'll modify it so I can explain better.