Are there any common applications where one can replace FFT with Quantum Fourier Transform?

One example where the QFT comes up is in simulating quantum particles' position and momenta. A typical use of the discrete Fourier Transform is classical applications is to convert time-domain signals into frequency-domain spectra, and an analogous relationship between the position and momentum domains allows you to convert a wavefunction that represents the (probability) distribution of a particle in space into a wavefunction that represents that same particle's momentum spectrum.

To study the dynamics of particles using a scheme like this, you have to specify a way to discretize position space so that you'll be able to represent a continuum of position/momentum values on a system with finite-dimensional wavefunctions (this involves defining the minimum/maximum position you will represent, the grid spacing in position/momentum space, and so on). An example of such a discretization or "digitization" formalism is given in [1], or for a more thorough review see [2].

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[1] Ronaldo D Somma. "Quantum simulations of one dimensional quantum systems". arXiv preprint: https://arxiv.org/abs/1503.06319

[2] Natalie Klco and Martin J. Savage. “Digitization of Scalar Fields for Quantum Computing.” Physical Review A 99.5 (2019): https://arxiv.org/abs/1808.10378