Hematite is a prototypical antiferromagnetic insulator for spintronic and magnonic applications, yet controlled studies of coherent magnons in thin films remain challenging due to strain sensitivity and spatial inhomogeneity. Here, we investigate magnon excitations in lithographically patterned hematite microdisks, and obtain reproducible magnon spectra from multiple individual microdisks. Brillouin light scattering reveals bulk-like field-dependent thermal magnon resonances with an ultralow effective damping parameter on the order of ~10-5. Coherent magnons are further excited using a stripline antenna adjacent to the microdisk, revealing a power-dependent frequency blue shift accompanied by a nonlinear evolution of the coherent response under strong microwave driving. Field-dependent analysis indicates a power-dependent increase of the effective magnetic anisotropy field, suggesting a tunable nonlinear response of antiferromagnetic magnons in laterally confined microdisks. These results establish patterned hematite thin films as a platform for exploring coherent and nonlinear antiferromagnetic magnon dynamics.