Let $E$ be a locally convex space, let $U$ be a convex balanced closed neighbourhood of $0$ in $E$. Let $F=\bigcap_{n\in\mathbf{N}}n^{-1}U$, which is a closed linear subspace of $E$. In fact $F$ is the kernel of the Minkowski semi-norm of $U$. This semi-norm induces a norm on the vector space $E/F$.
Is the described topology on $E/F$ the quotient topology?
The image of $U$ in the quotient topology is a neighbourhood of $0$, which does not contain a subspace. In order to prove the question we need to show that this image is bounded. So does any unbounded balanced convex neighbourhood of $0$ contain a subspace?
No. Let $E$ be $\ell^2$ with the norm topology, and $$U = \{x\in \ell^2 : \sum |x_n|^2/n^2 \le 1\}$$ Then $U$ is a convex closed neighborhood of $0$. Also, $F=\{0\}$ and $U$ induces a norm on $E/F = E$, namely $$ \|x\| = \sqrt{\sum |x_n|^2/n^2} $$ But this topology on $E$ is not the topology we started with. For example, the sequence of standard basis elements $\{e_n\}$ converges to $0$ with respect to $\|\cdot\|$, whereas it did not converge in the original topology of $E$.
Also, $U$ is an unbounded convex neighborhood of $0$ which does not contain any subspace. It can be described as an infinite-dimensional ellipsoid with semi-axes $1,2,3,4,\dots$