Can I use Arduino IDE with this STM32 MPU and Programmer?

From your pictures, looks like the programmer you have will not easily connect to your STM32 board. Also, your board appears to have a USB port attached. In my experience with other microcontroller breakout boards, the USB port is used to directly program the board.

See about gathering more information about your controller board. Especially look for information related to programming it.

Good luck with your project.