You certainly can retrofit your queen bedspread into a king size, or as you say, cut apart the queen bedspread and make more pillow shams.
I would opt to use the queen size bedspread for converting it into a king bedspread. You can go to the fabric store with the matching pillow and use it to match fabric to add a boarder onto the bedspread. Note, these alterations I am about to suggest are for a bedspread, not a comforter. A bedspread is a relatively thin cover for the bed. A comforter is a thicker, batting filled cover that is fluffier and has more depth than a bedspread. However, you can do the same with a comforter, but it is more difficult for one without alot of sewing experience.
There are two ways to add a boarder to a bedspread to make it larger. One way is to cut a 5" strip from the sides only. Then, add another fabric to those cut edges and then reattach the 5" you cut off back onto the added fabric. This causes a strip band on the sides of the bedspread to widen the width of the bedspread.
Adding a boarder directly on the edge of the bedspread is a matter of cutting fabric into wide strips, twice the width of the finished boarder. Fold the strips in half, lengthwise, iron. Then sew the strips to the side edges of the bedspread, or if you need length, also add to the length. That would be three edges of the bedspread to make it wider and longer.
With either way, you can bind off the raw edges on the bottom side if you have a serger or use double fold seam binding to encase the raw fabric edges.
A third way to deal with making a bigger bedspread is to cut the bedspread the size of the king mattress and then add additional fabric to three sides for the drop over the mattress and to cover the box spring.
The fabric you add to the bedspread can be a geometric, a stripe, a floral or a solid tone on tone--depending on what design you already have on the bedspread. It is best to use home decorating fabric rather than apparel fabric as the home dec fabric is more stable.
I hope these options help you decide what you want to do.