  Low-frequency 1/f noise in electronics is a conductance fluctuation, that has
been expressed in terms of a mobility "$\alpha$-noise" by Hooge and
Kleinpenning. Understanding this noise in graphene is a key towards
high-performance electronics. Early investigations in diffusive graphene have
pointed out a deviation from the standard Hooge formula, with a modified
expression where the free-carrier density is substituted by a constant density
$n_\Delta\sim10^{12}\;\mathrm{cm^{-2}}$. We investigate hBN-encapsulated
graphene transistors where high mobility gives rise to the non-linear
velocity-saturation regime. In this regime, the $\alpha$-noise is accounted for
by substituting conductance by differential conductance $G$, ressulting in a
bell-shape dependence of flicker noise with bias voltage $V$. The same analysis
holds at larger bias in the Zener regime, with two main differences: the first
one is a strong enhancement of the Hooge parameter reflecting the hundred-times
larger coupling of interband excitations to the hyperbolic phonon-polariton
(HPhP) modes of the mid-infrared Reststrahlen (RS) bands of hBN. The second is
an exponential suppression of this coupling at large fields, which we attribute
to decoherence effects. We also show that the HPhP bands control the amplitude
of flicker noise according to the graphene-hBN thermal coupling estimated with
microwave noise thermometry. The phenomenology of $\alpha$-noise in graphene
supports a quantum-coherent bremsstrahlung interpretation of flicker noise.
