Essentially, you're interested in moderation, which relates closely to interaction (I've added this tag to the OP). I found myself a little puzzled by the problem of applying it to a correlation rather than to a regression, but I've got an idea, and would welcome feedback or better alternative methods, if any.
Moderation analysis in multiple regression is straightforward, but requires choosing a dependent variable to predict (outcome), and produces slightly different results when you change your choices of outcome and predictors. Only bivariateregression produces the same effect-size estimate as a bivariate correlation regardless of which variable predicts which. If you add to these two, opposite-but- equal models only the product of your predictor variable and third variable, it seems you'll get slightly different effect size estimates, but identical ratios of these estimates to their standard-errors $(SE)$.
Hence if you want a simple, frequentist test of the statistical-significance of the null-hypothesis that water does not moderate the relationship between vegetation and temperature, you can test either model:
$$\text{vegetation}=a_1+\text{temperature}\times b_1+\text{temperature}\times\text{water}\times c_1$$
$$\text{temperature}=a_2+\text{vegetation}\times b_2+\text{vegetation}\times\text{water}\times c_2$$
where both $a=$ intercept coefficients, $b\approx$ bivariate correlation coefficients, and $c=$ the regression coefficient for water's moderating effect on $b$.
For a t-test of your evidence against $c=0$ (the null hypothesis I described above), $\frac{c}{SE_c}=t_{(df)}$ (two-tailed) with $df=N-3$ where $N=$ your number of observations and $3=$ the number of coefficients in the general-linear-model. The $p$ for that $t$ represents the probability that you would find a moderating effect of water at least as strong as in your data if you were to sample again randomly from the same population, and if there really is no moderating effect overall.
If the moderating effect in your data is strong, that probability should be quite small, which would support your alternative hypothesis. In this case, I'd also recommend calculating a confidence-interval (CI) around $c$ to give you a better sense of how reliable your estimate of that effect is. You can even skip the $t$-test entirely if you don't care so much about the precise probability of your sample given the null. The closest-to-zero boundary of a CI that excludes zero represents the value furthest from zero that you could have rejected as a null hypothesis at the level of $\alpha$ used in constructing the CI.
One bit that's still a little confusing to me about this is that $\frac{c_1}{SE_{c_1}}\ne\frac{c_2}{SE_{c_2}}$ if I center (cf. centering) the predictor variable before multiplying it by the moderator. Whether you center and which variable to choose as your predictor shouldn't make a tremendous difference (e.g., either way, you'll probably reach the same conclusion about the null hypothesis mentioned above), I think...but I'm not sure what to recommend in your case, because the issue strikes me as somewhat unresolved in general. For further reading on whether to center interacting predictors, see the following:
Also, if you're going to use this interaction term, it's important to include water as a predictor as well. Doing so may make your choice of predictor and outcome more important, and will effectively alter the question you're analyzing, but hopefully the change will be informative.
