I think this may be a simple question, but if we have two predictor variables where our regression model can be expressed by an equation of the form $$Y=\beta_0+\beta_1X_{t1}+\beta_2X_{t2}+\epsilon_t$$ and $\epsilon_t$ ~ $N(0,\sigma^2)$
When given $r(X_1,X_2)$ as our correlation, can we just say that the $VIF_1=VIF_2=\frac{1}{1-r^2(X_1,X_2)}$?
Yes. The only source of multicolinearity in your model the linear correlation between $X_1$, and $X_2$, and since in simple linear regression $R^2$ of the regression of $X_1$ on $X_2$ (or vice versa) equals the square of the Pearson correlation coefficient between $X_1$ and $X_2$, then indeed $VIF_1 = VIF_2$.