Whispering-gallery-mode (WGM) microresonators can greatly enhance the light-matter interaction, making them indispensable units for frequency conversion in nonlinear optics. Efficient nonlinear wave mixing in microresonators requires stringent simultaneous optical resonance and phase matching conditions. Thus, it is challenging to achieve efficient frequency conversion over a broad bandwidth. Here, we demonstrate broadband second-harmonic generation (SHG) in the x-cut thin-film lithium niobate (TFLN) microdisk with a quality factor above 107 by applying the cyclic quasi-phase matching (CQPM) mechanism, which is intrinsically applicable for broadband operation. Broadband SHG of continuous-wave laser with a maximum normalized conversion efficiency of ~15%/mW is achieved with a bandwidth spanning over 100 nm in the telecommunication band. Furthermore, broadband SHG of femtosecond laser, supercontinuum laser and amplified spontaneous emission in the telecommunication band is also experimentally observed. The work is beneficial for integrated nonlinear photonics devices like frequency converters and optical frequency comb generator based on second-order nonlinearity on the TFLN platform.