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.