We construct coherent states for each eigenspace of a magnetic Laplacian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini-Study Laplacian in a closed form involving of a terminating Kampe de Feriet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself.