As far as I'm aware, there are no ICs that perform just quadrature demodulation. If you did this, you'd no longer have an SDR as there would be no software, and having discarded the constraint of reprogrammability, there's usually a simpler hardware solution.
However, demodulation is commonly performed in FPGAs or DSPs in these situations, yielding the advantages of dedicated hardware (as opposed to performing it on a desktop computer), while retaining the programmable nature of SDRs. In the schematic you link, the very large IC in the middle is an FPGA. It's common in wideband SDRs like this for the FPGA to be responsible only for filtering some narrow band(s) out of the received signal and sending them, via ethernet, elsewhere for further processing. HPSDR and SDRstick are other SDRs intended to work this way.
But you could also add another DAC, audio amplifier, and speaker and program the FPGA to perform your desired demodulation directly. The radios by Elecraft are an example.