It's consistent that the answer is no. Bartoszynski and Judah prove the following on page 26 in their book, Set Theory: On the Structure of the Real Line:
Assume $MA_\kappa$. Then a partial order of size $\leq \kappa$ is ccc iff it is $\sigma$-centered.
Since a Cohen real adds a Suslin tree, there is a ccc iteration $Add(\omega) * \dot{T}$ of size $\aleph_1$ where $\dot{T}$ is forced to be a Suslin tree. Under $MA_{\aleph_1}$, this is $\sigma$-centered ($\omega$-centered), but the quotient forcing is not because Aronszajn trees are not $\sigma$-centered.
Mohammad was on the right track in the (deleted) comments. Maybe someone can construct a ZFC counterexample.