HDMI 5.1 output on old H61M-HVS

That 5.1 channel output spec is likely for the 3.5mm analog outputs on the board. For HDMI, it will most likely be using the Intel integrated GPU, which should support uncompressed multi-channel audio, probably up to 7.1 and depending on what options your AVR supports.
As pointed out, the sound chipset has no function over HDMI.
The soundcard is effectively what connects to the other end of HDMI, your receiver.
As long as your receiver isnt as old as an 8800GTX it should work.

Anything older might have HDCP handshake issues.
Worst case scenario, just drop in a GTX-1030, Actually, even a GT710 should work.