  Specific heat, $C_B$, of a TbAl$_3$(BO$_3$)$_4$ crystal was studied for 50 mK
$<T<$ 300 K, with emphasis on $T<1$ K, where a phase transition was found at
$T_c =0.68$ K. Nuclear, non-phonon ($C_m$), and lattice contributions to $C_B$
were separated. Lowering of $T_c$ with ncrease of magnetic field parallel to
the easy magnetization axis, $B_{||}$, was found. It was established that $C_m$
and a Gr\"uneisen 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 $0.82 \le z \le 0.96$ 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 $T_c$ with
increase in $B_{||}$ is opposite to the classical behavior. Hence, a dominant
role of QF was supposed.
