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We report, for the first time to our knowledge, the passively Q-switched laser performance of the Yb:YAl3(BO3)4 crystal operating at ~1 mum with a Cr4+:YAG crystal used as the saturable absorber. Employing a compact plano-concave resonator end pumped by a high-power diode, we obtained by using a 2 mm thick alpha -cut crystal a sigma-polarized average output power of 4.15 W at 1041.3 nm with optical-to-optical and slope efficiencies as high as 48.3% and 59%, respectively. The Q-switched output power generated with the c-cut crystal (2 mm thick) was lower due to the crystal inhomogeneity: 3.15 W with a slope efficiency of 48%. The highest pulse energy, shortest pulse duration and highest peak power achieved with the a- and c-cut crystals were found to be nearly identical, being 139 muJ, 18 ns, and 7.33 kW, respectively.