One should really only talk about $K_0$ of *small* categories, otherwise one runs into the difficulty explained in Matthias's comment. Assuming $\mathcal{B}$ is small, let me identify $\mathcal{A}$ with its essential image by $F$, which will be a (strictly) full triangulated category of $\mathcal{B}$. For a positive answer to your question, it is reasonable to impose the condition that $\mathcal{A}$ is *dense* in $\mathcal{B}$, i.e. every object of $\mathcal{B}$ is a direct summand of an object of $\mathcal{A}$. Then the claim is indeed true and follows for example from Thomason's classification theorem, which says that there is a bijection between dense strictly full triangulated subcategories $\mathcal{A} \subset \mathcal{B}$ and subgroups of the Grothendieck group $K_0(\mathcal{B})$. See Corllary 2.3 in [Thomason, *The classification of triangulated subcategories*, Comp. Math. 105 (1997), pp. 1-27].