We study the spontaneous scalarization of spherically symmetric, asymptotically flat boson stars in the $(alpha {cal R} + gamma {cal G}) phi^2$ scalar-tensor gravity model. These compact objects are made of a complex valued scalar field that has harmonic time dependence, while their space-time is static and they can reach densities and masses similar to that of supermassive black holes. We find that boson stars can be scalarized for both signs of the scalar-tensor coupling $alpha$ and $gamma$, respectively. This is, in particular, true for boson stars that are {it a priori} stable with respect to decay into individual bosonic particles. A fundamental difference between the $alpha$- and $gamma$-scalarization exists, though: while we find an interval in $alpha > 0$ for which boson stars can {it never} be scalarized when $gamma=0$, there is no restriction on $gamma eq 0$ when $alpha=0$. Typically, two branches of solutions exist that differ in the way the boson star gets scalarized: either the scalar field is maximal at the center of the star, or on a shell with finite radius which roughly corresponds to the outer radius of the boson star. We also demonstrate that the former solutions can be radially excited.
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