Solution:

Quadratic equations are the polynomial equations of degree $2$ in one variable of type $f(x) = ax² + bx + c$ where a, b, c, ∈ R and $a ≠ 0$. 

Let the two integers be $x$ and $x+1$, $x$ taken as the smaller integer.

From the question, the product of these two integers is $306$

So,

$x(x + 1) = 306$

⇒$ x² + x – 306 = 0$

Thus, the required quadratic equation is $ x² + x – 306 = 0$