This answer is wrong, the semion category has a nontrivial associator and so 1+X is not an algebra there. See Tobias's answer.

I think the answer to this question is "no." Below I explain a counterexample.

Consider the fusion category Vec(Z/2) with two objects 1 and X. The object 1+X has a natural Frobenius algebra structure (just think of it as the group ring C[Z/2]). However, Vec(Z/2) has two different *-structures: the usual *-structure where X is real (aka orthogonal) and the *-structure where X is pseudoreal (aka quaternionic aka symplectic). In the latter case 1+X can't have a Q-system structure by remark 3 on page 30 of Mueger's From Subfactors to Categories and Topology I which explains that Q-systems are always real.

The tricky point in the above is checking that Vec(Z/2) really does have a second *-structure. I worked this out diagrammatically, but it can also be realized by looking atU_q(sl_2) when q is a primitive 6th root of unity since spin 1/2 reps are pseudoreal. This tensor *-category is called the called the "semion" theory in Section 5.3.1. of Rowell-Strong-Wang's On classification of modular tensor categories where they note that the nontrivial object has "Frobenius-Schur indicator -1", in other words it's pseudoreal.