Is deciding if there's a solution to a single multivariate quadratic equation NP-hard?
For $\mathbb{R}$ (the real numbers), you can decide whether a single multivariate quadratic equation has any real roots or not, in polynomial time.  See https://cstheory.stackexchange.com/a/19858/5038.  There are some other answers to that question that partly address the case of other fields. 

For $\mathbb{F}_2$ (the integers modulo two), you can decide whether there is a root in polynomial time.  See https://cstheory.stackexchange.com/q/37687/5038.

I don't know what the situation is for $\mathbb{Q}$ or $\mathbb{F}_{p^n}$.