NIOS version that runs "count binary" template on Altera DE2-115 board

You can add PIO (Parallel I/O) megafunction to your QSYS and connect LEDs to NIOS this way. The value will be transferred to the FPGA pins and the LEDs will be turned on/off according calculation result. With a little modification of your C code and connection to the right FPGA pins (export interface from QSYS) it will work fine. For 7-segment display you need to write a bit more C code, to decode the value and turn on/off the needed segments of 7-segment elements. Unfortunatelly I am not familiar with this particular board and its LCD display.