  Spin nematic phase is a phase of frustrated quantum magnets with a
quadrupolar order of electron spins. Since the spin nematic order is usually
masked in experimentally accessible quantities, it is important to develop a
methodology for detecting the spin nematic order experimentally. In this paper
we propose a convenient method for detecting quasi-long-range spin nematic
correlations of a quadrupolar Tomonaga-Luttinger liquid state of $S=1/2$
frustrated ferromagnetic spin chain compounds, using electron spin resonance
(ESR). We focus on linewidth of a so-called paramagnetic resonance peak in ESR
absorption spectrum. We show that a characteristic angular dependence of the
linewidth on the direction of magnetic field arises in the spin nematic phase.
Measurments of the angular dependence give a signature of the quadrupolar
Tomonaga-Luttinger liquid state. In our method we change only the direction of
the magnetic field, keeping the magnitude of the magnetic field and the
temperature. Therefore, our method is advantageous for investigating the
one-dimensional quadrupolar liquid phase that usually occupies only a narrow
region of the phase diagram.
