Can I install USB 3.0 USB bracket and header if my motherboard does not support USB 3.0?

If the panel only needs a standard USB 3.0 header connection, then yes, you can buy an adapter to connect it to a USB 2.0 header. The result should be exactly the same as if connecting a 3.0 extension cable to a 2.0 port or vice versa – all of the 3.0 extra pins will remain unconnected, and devices will work in USB 2.0 mode.

However, the available power will also be limited to what USB 2.0 normally provides, and that might not be enough for all ports to work.

It would be better to buy a PCI-Express card with a USB 3.0 controller – some of them have an internal USB 3.0 pin header (or even two) in addition to the standard external ports.