You didn’t attach a file, but I suspect your quads are not planar. If that is the case, the unrolled edges will be slightly curved, which is what looks like is happening in you image. Perhaps you can post a sample file with the 3D object before unrolling so someone can verify…
Mitch was correct. The quad sides are warped and not planar so they unroll with curved edges. Note that if you try building this shape from sheet material the sides would need to be stretched or shrunk to go from flat to warped.
If you rebuild each quad side as two triangles with a common edge the shape should unroll without gaps.