Answer
Yes.
Work Step by Step
If the integer is even, write it as $2x$. Then, its square is $4x^2$ and it can be written as $4(x^2)$ so $x^2=m$. If the integer is odd, write it as $2y+1$. Then, the square is $4y^2+4y+1=4(y^2+y)+1$, so $m=y^2+y$ is valid.
