I've been given a region that is contained by a quadrilateral with vertices.
From this I need to find the 2-dimensional vector field $F = (M(x,y),N(x,y))$ such that
$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1$
I know I'll be using this and applying Green's Theorem later on but I don't know how to make use of the given quadrilateral to get a function that will allow me to apply the Theorem.
The only criteria you have given for $N$ and $M$ is that $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1$$ which is entirely independent of the shape you provided. You cannot find "the" vector field that satisfies this as there are infinitely many vector fields that do so. For example, any $M$ and $N$ of the form
$$\begin{align}N(x,y) & = f_1(y) & + ax & \;\;\;\;+ f_2(x) & - xg'_2(y)\\ M(x,y) & = g_1(x) & + (a-1)y & \;\;\;\; - yf'_2(x) & + g_2(y)\end{align}$$
where $a\in\mathbb{R}, $ $f_1$ and $g_1$ are any functions, and $f_2$ and $g_2$ are differentiable functions will meet the criteria that $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1 \; .$$