p is an odd prime.
Clearly Q represents all squares modulo p and all integers divisible by p, so the problem is really just the nonsquares modulo p. I started by letting $t$ be the smallest nonsquare mod p that is not represented by Q. I deduced that $t-1$ and $t-2$ must also be nonsquares, but I failed to derive a contradiction.
On the other hand legendre's three squares theorem says all integers can be represented by Q over the integers except those of the form $4^{n}(8k+7), n\in mathbb{N}, k\in mathbb{Z}$. But this doesn't mean numbers in that form cannot be represented in modulo p. For example $7\equiv 4^{2}+1+1 \mod[11]$. But I have no clue to prove that all numbers of the form $4^{n}(8k+7)$ are necessarily represented by Q modulo p.
As you note, the three squares theorem says (for example) that every integer of the form $8k+1$ is the sum of three squares.
On the other hand, for an odd prime $p$, it's easy to solve $8k+1\equiv t\pmod p$ for $k$, namely $k \equiv 8^{-1}(t-1) \pmod p$. Therefore any $t$ can be represented, modulo $p$, as the sum of three squares.