In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, including soliton, exponential and periodic solutions, which are useful to analyze the behaviors of the FSH equation. We show that the auxiliary equation method improves the speed of the discovery of exact solutions.