Pressure-dependent, low temperature inelastic light (Raman) scattering measurements of KCuF$_3$ show that applied pressure above $P^{*} sim$ 7 kbar suppresses a previously observed structural phase transition temperature to zero temperature in KCuF$_3$, resulting in the development of a $omegasim$ 0 fluctuational (quasielastic) response near $T sim$ 0 K. This pressure-induced fluctuational response --- which we associate with slow fluctuations of the CuF$_6$ octahedral orientation --- is temperature independent and exhibits a characteristic fluctuation rate that is much larger than the temperature, consistent with quantum fluctuations of the CuF$_6$ octahedra. A model of pseudospin-phonon coupling provides a qualitative description of both the temperature- and pressure-dependent evolution of the Raman spectra of KCuF$_3$.
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