Can the correlation coefficient of a quadratic model be calculated? I am very confused on the differences between linear and quadratic regression.
In linear regression, you have R, Pearson's correlation coefficient, which can tell you the strength and direction of a linear relationship, and R^2, the coefficient of determination, which can tell you how well the regression model fits the observed data, correct?
In quadratic regression, I believe it is slightly different. From my research, I have been unable to find out if R can be calculated for a quadratic model. If not, is there another kind of correlation coefficient that can describe the strength and direction of a quadratic relationship? I believe that in quadratic regression R^2 still acts as the coefficient of determination and can be used in the same fashion. Is this correct?

Remember what the sign of $R$ means.
$R>0$ means a generally increasing trend: as $x$ increases/decreases, $y$ tends to increase/decrease.
$R<0$ means a generally decreasing trend: as $x$ increases/decreases, $y$ tends to decrease/increase.
A quadratic polynomial includes increasing and decreasing sections. Consequently, it does not make sense to use $\pm$ the way that $R$ does.
However, $R^2$ makes sense. Fit the regression $\hat y=\hat\beta_0+\hat\beta_1x+\hat\beta_2x^2$ (drop the linear and/or intercept terms of you have compelling reasons for doing so, but probably only if you have compelling reasons for doing so), and calculate the $R^2$. Since this is linear in the parameters (even if you drop the intercept and/or linear term), the regression is linear, and $R^2$ has its usual interpretation of being the proportion of variance explained, so the strength of the relationship (bigger means a stronger relationship).
