I don't know how to solve this equation, really tried to Google it but Google foo is weak.
$$ \ m^{2} - n^{2} = 1 \\ (m-n)(m+n) = 1 \\ m-n = 1 \quad \& \quad m+n = 1 \\ ? $$
This is about as far as I can get. A reference to how to solve these problems would be great. Is this linear algebra or just basic algebra?
Your third line is not quite correct. If you're looking for integer solutions, then $m-n$ and $m+n$ will both be integers, and if a product of two integers is $1$, then the two factors must either both be $1$ or both be $\textit{-1}$ (since those are the only integers with integer inverses).
Other than that, you were almost there. Let's suppose in the first case that $m-n=m+n=1$. Then $m=1+n$ and from the second equation, $2n+1=1$, so $n=0$ and $m=1$. You can plug this back into the original equation to check that it is indeed a solution.
The case $m-n=m+n=-1$ is similar.
Your problem is similar to Pell's equation, which covers equations of the form $x^2-ay^2=1$, but in the case of Pell's equation $a$ is assumed to be non-square, otherwise the problem is quite easy as you've just seen.