Off delay timing function for a relay, using a Arduino
If you want a quick answer: yes you can.
But you must take into consideration some constraints:
In Off-Delay Timers we usually completely cut the power supply off the circuit during the off delay which can be up to 15 minutes and maybe more. So we must store some energy (usually in capacitors) to insure counting the time and changing the state of a relay. For that we use a bistable relay which is a relay that get a pulse to switch from one state to the other so it does not need to be continuously supplied to maintain its state.
You must study your system in order to have enough stored energy to count the off delay (which is relatively long) and to have enough energy to flip the bistable relay. For the microcontroller you must get it into sleep mode and wake up by an internal timer (usually many times due to the long time).