Signal to Quantization Noise Ratio (SQNR)
No need to do the convolution to find the output. Just use the fact that $z^n$ is an eigenfunction of the system with $z=e^{j\omega_0}$. This will lead to the following rule

$$x[n]=A\sin(\omega_0 n+\phi) \Rightarrow y[n]=|H(e^{j\omega_0})|A \sin(\omega_0 n+\phi+\angle H(e^{j\omega_0}))$$

That is, the output to a sinusoidal signal with angular frequency $\omega_0$ is a similar sinusoidal signal with the same angular frequency, but scaled by $|H(e^{j\omega_0})|$ and phase-shifted by $\angle H(e^{j\omega_0})$.

For SQNR calculation, we only need the scaling $|H(e^{j\omega_0})|$. You probably know how to find the power of a sinusoidal signal, given its amplitude. Right?

Then you need to know how to calculate SQNR when a sinusoidal signal with amplitude $|H(e^{j\omega_0})|A$ gets quantized.