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.

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.