This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below.
Let $A_0$ be a finitely generated universal unital complex algebra $A_0=\langle \mathcal{G}\mid\mathcal{R} \rangle$. If an algebra $B$ has $|\mathcal{G}|$ generators which also satisfy the relations $\mathcal{R}$, then there exists a surjective algebra homomorphism $\pi:A_0\to B$.
If $\mathcal{R}$ consists of relations palindromic in the generators (and no complex numbers), then the complex conjugate opposite algebra $\overline{A_0^{\text{opp}}}$ satisfies the same relations it is the case that the induced homomorphism $r:A_0\to \overline{A_0^{\text{opp}}}$ is an algebra homomorphism. In this case I think we have $A_0= \overline{A_0^{\text{opp}}}$ as sets, and using this identification, $r:A_0\to A_0$, a conjugate linear map: $$r(g_1g_2)=g_2g_1\qquad (g_1,g_2\in \mathcal{G}),$$ is in fact an involution, a star. Equipping $A_0$ with this involution, call it $A$, this is then a *-algebra.
Does this mean we can equip an algebra quotient $\pi:A_0\to B$ with a star structure? Say $$\pi(a)^*=\pi(r(a))\qquad (a\in A).$$