Let $g: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for all $m, n \in \mathbb{N}$, the expression

$$\left(g(m) + n\right)\left(g(n) + m\right)$$

is a perfect square. Our goal is to find all such functions $g$.

### Step 1: Analysis of the Condition

Consider specific values of $m$ and $n$. Setting $m = n$, we get:

$$\left(g(m) + m\right)\left(g(m) + m\right) = \left(g(m) + m\right)^2$$

Clearly, this is a perfect square by construction of the square $\left(g(m) + m\right)^2$.

### Step 2: Exploring Generality

Now, consider $m \neq n$. The condition is:

$$\left(g(m) + n\right)\left(g(n) + m\right) = k^2$$

for some integer $k$. A productive approach is to try simple forms for $g$.

### Step 3: Choosing a Function Form

Suppose $g(n) = n + c$ for some constant $c \in \mathbb{Z}_{\ge 0}$. Let's verify this form:

1. Substitute into the condition:
$$\left((n + c) + m\right)\left((m + c) + n\right) = (n + m + c)(m + n + c)$$

2. Simplifying, we have:
$$= (n + m + c)^2$$
which is clearly a perfect square since it is the square of $(n + m + c)$.

### Step 4: Verification and Generalization

We found that $g(n) = n + c$ satisfies the condition for \emph{any} non-negative integer $c$.

### Conclusion

Therefore, the functions of the form $g(n) = n + c$ where $c \in \mathbb{Z}_{\ge 0}$ are indeed all possible solutions that satisfy the given condition that the expression is a perfect square for all $m, n \in \mathbb{N}$.

The complete set of functions is:

$$g(n) = n + c \quad \text{for some} \quad c \in \mathbb{Z}_{\ge 0}.$$

Thus, the solution to the problem is:
$$\boxed{g(n) = n + c \text{ for some } c \in \mathbb{Z}_{\ge 0}}.$$