$\dim_H J(f_c) \ge 1$ for $c \in M$ by connectedness and uncountability of $J(f_c)$. For which points is there equality? $c=0$ and $c=-2$ for sure, but is this an exhaustive list?
Notation: $\dim_H$ is Hausdorff dimension, $J$ is Julia set, $f_c(z) = z^2 + c$, $M = \{ c : J(f_c) \text{ is connected}\} \subset \mathbb{C}$ is the Mandelbrot set.
It follows from a theorem due to Zdunik: https://link.springer.com/content/pdf/10.1007/BF01234434.pdf
(theorem 2).
This theorem states that if $f$ is a rational map, and $m$ is its measure of maximal entropy, then the dimension of the Julia set is strictly larger than the dimension of $m$ unless $f$ has a parabolic orbifold.
For a polynomial with connected Julia set (in particular for a quadratic polynomial in the Mandebrot set), it is known that the dimension of $m$ is 1 (this is due to Makarov).
It is too complicated to explain here what it means for $f$ to have a parabolic orbifold, but suffice it to say that the only quadratic polynomials with that property are (up to affine conjugacy) $z^2$ and $z^2-2$.