Quantum systems can be dynamically controlled using time-periodic external fields, leading to the concept of Floquet engineering, with promising technological applications. Computing Floquet band structures is harder than only computing ground state properties or single time-dependent trajectories, and scales exponentially with the Hilbert space dimension. Especially for strongly correlated systems in the low frequency limit, classical approaches based on truncation break down. Here, we present two quantum algorithms to determine effective Floquet modes and band structures. We combine the defining properties of Floquet modes in time and frequency domains with the expressiveness of parameterized quantum circuits to overcome the limitations of classical approaches. We benchmark our algorithms and provide an analysis of the key properties relevant for near-term quantum hardware.