Your question seems to contain two parts.
First, you're asking how to set up the equations of motion for this coupled system.
Second, you are asking how to use symmetry considerations to find the normal modes and frequencies.
Let's first answer the bit about how to find the equations of motion, because it seems like you already have some idea of how to deal with the symmetry part.

Eqns of motion
===

Let the left circuit be called #1 and the right circuit be called #2.
Denote the inductance and capacitance of circuit #1 as $L_1$ and $C_1$, and similarly for circuit #2.

Denote by $V_1$ and $I_1$ the voltage across and current through $L_1$, and similarly for $V_2$ and $I_2$.

A mutual inductance means that the two inductors share flux.
In particular, the flux in $L_1$ due to the current $I_2$ is

$$\Phi_{1,2} = M I_2$$.

This is just the definition of mutual inductance.
This flux *adds* to the self flux $\Phi_{1,1}$ of circuit #1.
Therefore, we have

$$\Phi_1 = \Phi_{1,1} + \Phi_{1,2} = L_1 I_1 + M I_2$$

Differentiate both sides to get

$$\dot{\Phi}_1 = - L_1 \ddot{Q}_1 - M \ddot{Q}_2 $$

where here we used $I_1 = -Q_1$ which is correct because current flowing downward through each inductor is flowing *away* from the corresponding capacitor.
You learned that the time rate of change in flux of an inductor gives the voltage across it: $\dot{\Phi} = V$.
Using this fact gives us

$$V_1 = -L_1 \ddot{Q}_1 - M \ddot{Q}_2 . $$

The voltage across the inductor and capacitor are the same because they're wired in parallel. Therefore, $V_1 = Q_1 / C_1$ by the definition of capacitance, resulting in

$$
\begin{align}
\omega_1^2 Q_1 &= - \ddot{Q}_1 - (M/L_1) \ddot{Q}_2 \\
\text{and by symmetry} \qquad
\omega_2^2 Q_2 &= - \ddot{Q}_2 - (M/L_2) \ddot{Q}_1
\end{align}
$$

where $\omega_i \equiv 1/\sqrt{L_i C_i}$.
Are these equations reasonable?
If $M=0$ then we have
$$\omega_1^2 Q_1 = -\ddot{Q}_1 \qquad \text{and} \qquad \omega_2^2 Q_2 = - \ddot{Q}_2$$

which is precisely what you expect for uncoupled oscillators.
Therefore, our equations of motion are probably correct.

So now you know how to get the equations of motion. Can you take it from here?