Specific heat, CB, of a TbAl3(BO3)4 crystal was studied for 50 mK < T < 300 K, with emphasis on T < 1 K, where a phase transition was found at Tc = 0.68 K. Nuclear, non-phonon (Cm), and lattice contributions to CB were separated. Lowering of Tc with ncrease of magnetic field parallel to the easy magnetization axis, B || , was found. It was established that Cm and a Grüneisen ratio depend on B || and T in a way characteristic of systems, in which a classical transition is driven by quantum fluctuations, QF, to a quantum critical point at T = 0, by tuning a control parameter (B || ). The B || − T phase diagram was constructed and the dynamical critical exponent 0.82 ≤ z ≤ 0.96 was assessed. Nature of the transition was not established explicitly. Magnetization studies point at the ferromagnetic ordering of Tb 3+ magnetic moments, however, lowering of Tc with increase in B || is opposite to the classical behavior. Hence, a dominant role of QF was supposed.