I know that data from accumulator register is always transferred to input A register of ALU and processes to add or subtract by one as it is to be incremented or decremented and then is transferred back to accumulator register.

Do X Index and Y Index registers do the same to use ALU for increment or decrement?

Bryan

Yes. Except for the Program Counter, which has it's own dedicated increment circuitry, everything else goes through ALU, including index register increment/decrement, address calculations and stack pointer inc/dec.