USB mouse with a PS/2 adapter for FPGA PS/2 interface
This absolutely depends on the mouse. I have a USB Microsoft mouse that comes with a similar USB to PS2 adapter. The mouse can sense the difference and reconfigure itself to produce the different protocol but the mouse is marked as such.

99% of the time these adapters are wiring only - there is no intelligence whatsoever within them.

Two things to bear in mind - PS/2 is open collector and the data/clock voltages are 5V. I've been suspicious in the past that these USB/PS2 mice are putting 3.3V on the lines and they get away with it - something to check with a scope!

Edit: Related question which might help you in some way: How to write VHDL program to send command to PS2 device