159716

Because the question used a quarter and a third, and you can't have parts of coins, I thought of the smallest possible number that 4 and 3 would go into without a remainder.
I thought of 12, so one quarter of 12 is heads, that's 3 coins, but it didn't work because once I'd turned over 2 then 2+3= 5 coins being heads which is not one third of 12.
The next number is 24. A quarter of 24 is 6 coins being heads at the start, after turning 2 over that makes 6+2= 8 coins being heads. 8 is a third of 24, so it works.