Minimum value

If n and m are consecutive positive integers, then m = n - 1 (if we take the inequality into account, then we can see that n > m).

$n^2 - m^2 > 20$

$n^2 - (n - 1)^2 > 20$

$2n - 1> 20$

$n > 10.5$

$n ≥ 11$

$m ≥ 10$

Hence, the minimum value is:

$11^2 + 10^2 = \boxed {221}$