Let first number be $x$, and the second number be $y.$
$(x^{2} - y^{2}) = 45$
$\implies (x - y)(x + y) = 45$
The factors of $45$ are $15, 3, 9, 5, 1$ and $45$
Hence, the possible pairs of numbers are $(9,6), (7,2)$ and $(23,22).$
$\therefore$ The number of such pairs $= 3.$
The correct answer is $(C).$