Let $z_0$ be a fixed point of $f_c$. By the definition of the multiplier, $$f_c^\prime(z_0)=2z_0=\lambda$$

This implies $z_0=5e^{\pi i/4}/12$.

Since $z_0$ is a fixed point of $f_c$, $$f_c(z_0)=z_0^2+c=z_0$$

Thus, $c=5e^{\pi i/4}/12-25e^{\pi i/2}/144$.

This $c$ value gives the Julia set

The white dots correspond to $-0.723234 - 0.561024 i$, $0.502947 + 0.932521 i$, and $-0.322011 + 1.05903 i$, which form an orbit of period $3$.

The red dots correspond to $-0.858598 + 0.631395 i$, $0.633157 - 0.963212 i$, and $-0.232262 - 1.09871 i$, which form an orbit of period $3$.

The green dot is located at $0.705372 - 0.294628 i$ and the pink dot is located at $0.294628 + 0.294628 i$. These are both fixed points of $f_c$, $c=5e^{\pi i/4}/12-25e^{\pi i/2}/144$.