The product of two consecutive positive integers is 306.

Let the two consecutive positive integers be x and (x + 1).

According to the given condition,

$x(x+1)=306$

$\Rightarrow x^{2}+x-306=0$

$\Rightarrow x^{2}+18 x-17 x-306=0$

$\Rightarrow x(x+18)-17(x+18)=0$

$\Rightarrow(x+18)(x-17)=0$

$\Rightarrow x+18=0$ or $x-17=0$

$\Rightarrow x=-18$ or $x=17$

∴ x = 17           (x is a positive integer)

When x = 17,
x + 1 = 17 + 1 = 18

Hence, the required integers are 17 and 18.