Every \addlegendentry command adds an entry to the legend starting from the first \addplot command. It is irrelevant where you place the \addlegendentry command (it must be added after the relevant \addplot command, of course). Thus, with your code you add two legend entries that refer to the first and the second \addplot command respectively and not to the last two. Since you did not provide the option area legend to the first two \addplot commands (which is totally correct, because you don't want to add a legend entry for these plots anyways), you will get (dashed or thick) lines as legend picture.
You can verify this behaviour by adding some more \addlegendentry commands to your code. You will see that the legend pictures will have the same styling as the respective \addplot macros:
So, what you essentially need to do is to skip the first three \addplot commands so that legend entries will only be added for the last two \addplot commands. You can use the option forget plot to achieve this (see also this answer).
Having added the option forget plot to the first three \addplot commands, the correct legend images will magically appear:
\documentclass[12pt]{article}
\usepackage{pgfplots}
\pgfplotsset{width=9cm, compat=1.10}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
title = {$\delta = 0.9$},
xlabel = {$\xi$},
ylabel = {$c(\xi)$},
xmin = 0,
xmax = 1,
ymin = 0.17,
ymax= 1,
legend pos = south east,
]
\addplot[
name path = A,
domain= 0:1,
samples= 100,
line width = 1.2 pt,
forget plot
]
{(1+x)/2};
\node at (axis cs: .8, .84){\footnotesize$C(\xi)$};
\addplot[
name path = B,
domain= 0:1,
samples= 100,
style = dashed,
forget plot
]
{(1/4 + (0.9 +x)/2};
\node at (axis cs: 0.2,0.87){\footnotesize$\bar{c}(\xi;\delta)$};
\addplot[
name path = C,
domain= 0:1,
samples= 100,
style = dashed,
forget plot
]
{(3/4 + (-0.9 +x)/2};
\node at (axis cs: .8, .63){\footnotesize$\underline{c}(\xi;\delta)$};
\addplot[fill=green, fill opacity = 0.3, draw = none, area legend] fill between [of = A and C, soft clip={domain=0:1}];
\addplot[fill=red, fill opacity = 0.3, draw = none, area legend] fill between [of = B and A, soft clip={domain=0:1}];
\addlegendentry{$Q_{ABS} < Q_{S,PC}$}
\addlegendentry{$Q_{ABS} > Q_{U,PC}$}
\end{axis}
\end{tikzpicture}
\label{fig:SustProf}
\caption{Some Caption}
\end{figure}
\end{document}
In certain cases, forget plot might lead to unwanted side effects, because it does not only remove the plot from the legend. You can also put empty legend entries instead using \legend{,,,$Q_{ABS} < Q_{S,PC}$,$Q_{ABS} > Q_{U,PC}$};:
\documentclass[12pt]{article}
\usepackage{pgfplots}
\pgfplotsset{width=9cm, compat=1.10}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
title = {$\delta = 0.9$},
xlabel = {$\xi$},
ylabel = {$c(\xi)$},
xmin = 0,
xmax = 1,
ymin = 0.17,
ymax= 1,
legend pos = south east,
]
\addplot[
name path = A,
domain= 0:1,
samples= 100,
line width = 1.2 pt,
]
{(1+x)/2};
\node at (axis cs: .8, .84){\footnotesize$C(\xi)$};
\addplot[
name path = B,
domain= 0:1,
samples= 100,
style = dashed,
]
{(1/4 + (0.9 +x)/2};
\node at (axis cs: 0.2,0.87){\footnotesize$\bar{c}(\xi;\delta)$};
\addplot[
name path = C,
domain= 0:1,
samples= 100,
style = dashed,
]
{(3/4 + (-0.9 +x)/2};
\node at (axis cs: .8, .63){\footnotesize$\underline{c}(\xi;\delta)$};
\addplot[fill=green, fill opacity = 0.3, draw = none, area legend] fill between [of = A and C, soft clip={domain=0:1}];
\addplot[fill=red, fill opacity = 0.3, draw = none, area legend] fill between [of = B and A, soft clip={domain=0:1}];
\legend{,,,$Q_{ABS} < Q_{S,PC}$,$Q_{ABS} > Q_{U,PC}$};
\end{axis}
\end{tikzpicture}
\label{fig:SustProf}
\caption{Some Caption}
\end{figure}
\end{document}
The output is the same as above.
