There may be a way to get the fit coefficients out of the ggplot() output but I would just directly calculate them.
FIT <- Infil_Data2 %>% group_by(Site_ID) %>%
do(tidy(lm(Cal_Vol_cm ~ I(Sqrt_Time.x^2) + Sqrt_Time.x, data = .)))
> FIT
# A tibble: 9 x 6
# Groups: Site_ID [3]
Site_ID term estimate std.error statistic p.value
<chr> <chr> <dbl> <dbl> <dbl> <dbl>
1 H1 (Intercept) 0.0240 0.0478 0.501 0.630
2 H1 I(Sqrt_Time.x^2) 0.00854 0.000557 15.3 0.000000327
3 H1 Sqrt_Time.x -0.0322 0.0106 -3.04 0.0161
4 H2 (Intercept) -0.0577 0.359 -0.161 0.876
5 H2 I(Sqrt_Time.x^2) 0.0154 0.00418 3.68 0.00618
6 H2 Sqrt_Time.x 0.0516 0.0796 0.648 0.535
7 H3 (Intercept) 0.0235 0.0553 0.425 0.682
8 H3 I(Sqrt_Time.x^2) 0.00839 0.000644 13.0 0.00000114
9 H3 Sqrt_Time.x -0.00801 0.0123 -0.653 0.532
I cribbed that method of calculation from Use dplyr to do grouped t-tests and get number of observations simultanously