Thanks a lot for the detailed troubleshooting instructions. I can connect to either a thunderbolt port or, e.g., a display port, just not both at the same time. When I do, it either fails to recognize the second display, or sees both displays as a single extended display (only one shows up in the display menu or system report, but the image is spread across both). The USB tab does show that the dock is connected, and the name of the host driver is "AppleUSBXHCITR" (I assume that's the "Titan Ridge XHCI" you mentioned). I've tried a dozen or so different cables; so, I'm sure that's not it. The only thing left to try is a firmware update. Unfortunately, I'll have to wait until I can get my hands on a Windows device to do that.
Thanks again for the help. If I can't get the HP G2 to work, I'll look at newer TB4 docks.
To be clear, you are using a new MacBook Pro with M1 Pro CPU which supports two external displays?
Not an earlier Mac with M1 processor that supports only one external display?
The M1 Pro only supports two external displays so make sure no other display is connected to HDMI or Thunderbolt.
You didn't say if the connection to the dock is 40 Gbps x1 link width x2 (see the Thunderbolt tab in System Information.app).
You say image is spread across both. Do you mean they are mirrored (have the exact same contents)? That would only happen if you are connecting both displays to the MST hub of the HP Thunderbolt Dock G2.
Looking at the rear of the HP Thunderbolt Dock G2, the ports on the bottom from left to right are:
1) SS USB-C With DisplayPort Alt Mode (DP) (connecting a VGA display disables DisplayPort Alt Mode capability but the port can still be used for USB devices)
2) DisplayPort 1 (supports DisplayPort Dual Mode DP++)
3) DisplayPort 2 (supports DisplayPort Dual Mode DP++)
4) Thunderbolt 3.
One display needs to be connected to the Thunderbolt port (rightmost port). The other display can be connected to one of the three other ports (those other three ports are from the built-in DisplayPort 1.4 MST hub of the dock).