Multiple monitors without DisplayLink?

The other thing to do is to check System Settings > Displays to see if this is just a matter of changing preferences for whether the displays should be extended or mirrored.

Extend or mirror your Mac desktop across multiple displays - Apple Support

To drive two displays from a single Mac host port, that port must be a Thunderbolt port, and the device plugged into it must be a Thunderbolt device (such as a Thunderbolt hub or dock).

Is this the dock in question?

https://www.hp.com/us-en/monitors-accessories/computer-accessories/thunderbolt-G4-dock.html

Or maybe this one? (Same one or similar?)

https://www.hp.com/us-en/shop/pdp/hp-thunderbolt-dock-280w-g4-dock-w-combo-cable

If so,

• Are you connecting it to the Mac using a Thunderbolt 4 cable – not just a USB-C one?
• Are you connecting the cable to the "USB Type-C™ Thunderbolt 4 Port" on the dock – not the "USB Type-C™ Alt Mode Port (Video, Data, Power)" one? Both ports have USB-C connectors, but if you plugged the uplink cable to the computer into the wrong port, you might not be able to make a Thunderbolt connection if you could make any useful connection at all.

I also note that this dock has FOUR video output ports - two DisplayPorts, a HDMI port, and a USB-C port which can carry video. Macs do not support DisplayPort MST daisy-chaining, and if this dock is using the equivalent of that to provide signals on FOUR ports, it may be that there are pairs of ports that, on a Mac, won't see independent signals. There may be some combinations of two ports that work and others that don't. It may take experimenting or going carefully through the dock's manual to figure out which are which.