Small grid-connected inverters are not friendly to the electrical grid, in the sense they do not take any action to support the grid when contingency events occur. For example, because of their relatively low power capacity, small grid-connected inverters are not designed to provide dynamic frequency support to the grid. On the other hand, it is well known that microgrids and weak grids including distributed generation would benefit significantly if all of the grid-connected converters could support and help against grid frequency disturbances. Within the family of small grid-connected converters, single-phase quasi-Z-source inverters (QZSI) have become an attractive topology, because they represent a reliable and economical alternative, and can be very efficient in applications that demand small or medium powers. However, a major disadvantage is that the control strategy must manage both direct current and alternating current variables through the same group of switches. The latter is a challenging task when implementing predictive control schemes. This paper proposes a finite control set model predictive control (FCS-MPC) strategy for a single-phase grid-supporting QZSI. The proposed predictive scheme can be easily integrated with a complementary control block to provide grid frequency support. Experimental results show evidence of the inverter operating under the proposed control strategy and providing grid frequency support, which demonstrates the feasibility of the proposal.