Unfortunately SolidWorks doesn't do UV mapping. If you want to UV unwrap your model, you will have to do so in a program like Blender.
Yes, as Eric mentioned, you're better off using a polygonal modeling program like Blender/3dsMax/Maya/C4D etc, Blender being Free and having tons of tutorials online, that would be your best bet in a pinch. It's not the same as modeling in Solidworks, but you'll closer to the result you're seeking with proper UVs.
That being said, you would still need to have a 'seam' somewhere, imagine your surface is paper, you can't get away with having those 90 degree folds without having seams on your corners, but with Blender you can at least define where those seams are.