Is pair $(m, -m)$ which $m\in \mathbb{Z}$ is isomorphic to $\mathbb{Z}$? Is $\{(m, -m)\mid m\in \mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$?
What i know about isomorphic is, simply it is a bijective homomorphism. 
How one can define isomorphism map $\phi$ in above case?

The pair $(m, -m)$ is an element of $\mathbb Z^2$
$\mathbb Z$ is a set, and with the operation $+$ it is a group.

Speaking about homomorphisms between the two makes very little sensem because $(m, -m)$ is not a group (because even though technically, it is a set of two elements, you haven't defined an operation on it).

What I feel you are really trying to ask is the following question:

Is the set $\{(m, -m)| m\in\mathbb Z\}$ isomorphic to $\mathbb Z$?

In which case the answer is still unclear, because you still haven't defined an operation. Now, if the operation is $$(m, -m) + (n, -n) = (m+n, -(m+n)),$$
then yes, the two groups are isomorphic, and one possible isomorphism is the most obvious mapping you can possibly think of.
