Qualcomm already designs its own GPU IP, but it is targeted specifically at mobile. There would be significant additional work required to scale this technology up from mobile to high performance desktop, but if Qualcomm thought it would have a reasonable chance of success, there's really nothing stopping them from going for it.
However, the market is fairly small, and the R&D intensity is quite high, so it's not clear to me if it would be worth it for Qualcomm, or some other company with mobile GPU IP (ARM, Imagination Tech, Intel, etc.) to jump into the ring.
qualcomm's GPU is/was ATI's IP if I'm not mistaken.