This statement is true for some integers but false for others.
Work Step by Step
Let $m=1$ and $n=1$. Then $(m+n)/2=(1+1)/2=1$, so this is an example of the average of odd integers being odd. But now let $m=1$ and $n=3$. Then $(m+n)/2$$=(1+3)/2=4/2=2$. Since this is an example of two odd integers having an even average, it serves as a counterexample.