We present the first measurement of the gravitational quadrupole moment of the companion star of a spider pulsar, namely the black widow PSR J2051-0827. To this end, we have re-analysed radio timing data using a new model that is able to account for periastron precession caused by tidal and centrifugal deformations of the star as well as by general relativity. The model allows for a time-varying component of the quadrupole moment, thus self-consistently accounting for the ill-understood orbital period variations observed in these systems. Our analysis results in the first detection of orbital precession in a spider system at $\dot{\omega } = -68{_{.}^{\circ}}6_{-0{_{.}^{\circ}}5}^{+0{_{.}^{\circ}}9}$ yr-1 and the most accurate determination of orbital eccentricity for PSR J2051-0827 with e = (4.2 ± 0.1) × 10-5. We show that the variable quadrupole component is about 100 times smaller than the average quadrupole moment $\bar{Q} = -2.2_{-1}^{+0.6} \times 10^{41} \ {\rm kg\,m^2}$ . We discuss how accurate modelling of high-precision optical light curves of the companion star will allow its apsidal motion constant to be derived from our results.