From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into the tens column. From the tens column, we see that $1 + 6 + p + 8$ must have a ones digit of 4. Since $p$ is between 1 and 9, inclusive, then $1 + 6 + p + 8$ is between 16 and 24. Since its ones digit is 4, then $1 + 6 + p + 8 = 24$ and so $p = 9$. This also means that there is a carry of 2 into the hundreds column. From the hundreds column, we see that $2 + n + 7 + 5$ must have a ones digit of 0. Since $n$ is between 1 and 9, inclusive, then $2 + n + 7 + 5$ is between 15 and 23. Since its ones digit is 0, then $2 + n + 7 + 5 = 20$ and so $n = 6$. This also means that there is a carry of 2 into the thousands column. This means that $m = 2$. Thus, we have $m + n + p + q = 2 + 6 + 9 + 7 = 24$.