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

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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][1] 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][2] where they note that the nontrivial object has "Frobenius-Schur indicator -1", in other words it's pseudoreal.


  [1]: http://arxiv.org/abs/math/0111204
  [2]: http://arxiv.org/abs/0712.1377