Can you have different clocks working at different intervals to operate a flip flop?
Yes, you can do it with a J-K flip-flop. Look at the logic table. You can play with it in different manners. Maybe you won't even need a XOR gate. But you may need an OR gate to connect the two clock lines to a single CLK input. If the clock is also feeding the inputs, then you must add a small RC delay circuit (1Kohm + 100pF is enough) to make sure the signal arrives at the J and/or K inputs before it arrives at the CLK input.

Also make sure that the two clock pulses are distinct. Not rising at too close interval or practically at the same time. The interval must at least twice the total propagation delay of  the whole component chain. 1µs should be ok.