In this paper we dynamically determine the quadrupole mass moment Q of the magnetic white dwarf WD 0137-349 by looking for deviations from the third Kepler law induced by Q in the orbital period of the recently discovered brown dwarf moving around it in a close 2-hr orbit . It turns out that a purely Newtonian model for the orbit of WD 0137-349B , assumed circular and equatorial , is adequate , given the present-day accuracy in knowing the orbital parameters of such a binary system . Our result is Q = ( -1.5 \pm 0.9 ) \times 10 ^ { 47 } kg m ^ { 2 } for i = 35 deg . It is able to accommodate the 3-sigma significant discrepancy of ( 1.0 \pm 0.3 ) \times 10 ^ { -8 } s ^ { -2 } between the inverse square of the phenomenologically determined orbital period and the inverse square of the calculated Keplerian one . The impact of i , for which an interval \Delta i of possible values close to 35 deg is considered , is investigated as well .