Solution:

Let us say, the two consecutive positive integers be x and x + 1.

Therefore, as per the given questions,

${{x}^{2}}~+\text{ }{{\left( x~+\text{ }1 \right)}^{2}}~=\text{ }365$

$\Rightarrow ~{{x}^{2~}}+~{{x}^{2~}}+\text{ }1\text{ }+\text{ }2x~=\text{ }365$

$\Rightarrow 2{{x}^{2}}~+\text{ }2x-\text{ }364\text{ }=\text{ }0$

$\Rightarrow ~{{x}^{2~}}+~x-182\text{ }=\text{ }0$

$\Rightarrow ~{{x}^{2~}}+\text{ }14x-13x-182\text{ }=\text{ }0$

$\Rightarrow ~x\left( x~+~14 \right)\text{ }-13\left( x~+~14 \right)\text{ }=\text{ }0$

$\Rightarrow \left( x~+\text{ }14 \right)\left( x-13 \right)\text{ }=\text{ }0$

Thus, either, $x~+\text{ }14\text{ }=\text{ }0\text{ }or~x-13\text{ }=\text{ }0$ ,

$\Rightarrow ~x~=\text{ }\text{ }-14\text{ }or~x~=\text{ }13$

since, the integers are positive, so x can be 13, only.

$\therefore ~x~+\text{ }1\text{ }=\text{ }13\text{ }+\text{ }1\text{ }=\text{ }14$

Therefore, two consecutive positive integers will be 13 and 14.