= Solution
The specified <energy> requires $n_x+n_y=2$, so the relevant <energy eigenspace> has the <orthonormal basis>
$$
|2,0\rangle,\quad |1,1\rangle,\quad |0,2\rangle.
$$
Applying the <creation and annihilation operators> gives
$$
\begin{aligned}
\hat Q|2,0\rangle&=i\sqrt2\hbar|1,1\rangle,\\
\hat Q|1,1\rangle&=-i\sqrt2\hbar|2,0\rangle+i\sqrt2\hbar|0,2\rangle,\\
\hat Q|0,2\rangle&=-i\sqrt2\hbar|1,1\rangle.
\end{aligned}
$$
Thus its <matrix> in this ordered <basis> is
$$
Q=\hbar\begin{pmatrix}
0&-i\sqrt2&0\\
i\sqrt2&0&-i\sqrt2\\
0&i\sqrt2&0
\end{pmatrix}.
$$
Writing $q=\hbar\lambda$, the <characteristic polynomial> gives $\lambda(\lambda^2-4)=0$. The possible <angular momentum> <eigenvalues> are therefore
$$
\boxed{q=-2\hbar,\quad0,\quad+2\hbar.}
$$
For completeness, normalized simultaneous <energy> and <angular momentum> <eigenstates> are
$$
\begin{aligned}
|q=+2\hbar\rangle&=\tfrac12\bigl(|2,0\rangle+i\sqrt2|1,1\rangle-|0,2\rangle\bigr),\\
|q=0\rangle&=\tfrac1{\sqrt2}\bigl(|2,0\rangle+|0,2\rangle\bigr),\\
|q=-2\hbar\rangle&=\tfrac12\bigl(|2,0\rangle-i\sqrt2|1,1\rangle-|0,2\rangle\bigr).
\end{aligned}
$$
Each value occurs once in this <energy eigenspace>. Equivalently, <circular ladder operators of a two-dimensional oscillator> give $\hat H=\hbar\omega(\hat N_++\hat N_-+1)$ and $\hat L_z=\hbar(\hat N_+-\hat N_-)$. With $N_++N_-=2$, the difference can only be $2,0,-2$.
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