We report the results of a Suzaku X-ray imaging study of NGC 6251 , a nearby giant radio galaxy with intermediate FR I/II radio properties . Our pointing direction was centered on the \gamma -ray emission peak recently discovered with Fermi -LAT around the position of the north-west radio lobe 15 arcmin offset from the nucleus . After subtracting two “ off-source ” pointings adjacent to the radio lobe , and removing possible contaminants in the XIS field of view , we found significant residual X-ray emission most likely diffuse in nature . The spectrum of the excess X-ray emission is well fit by a power law with photon index \Gamma = 1.90 \pm 0.15 and a 0.5 - 8 keV flux of 4 \times 10 ^ { -13 } erg cm ^ { -2 } s ^ { -1 } . We interpret this diffuse X-ray emission component as being due to inverse-Compton up-scattering of the cosmic microwave background photons by ultrarelativistic electrons within the lobe , with only a minor contribution from the beamed emission of the large-scale jet . Utilizing archival radio data for the source , we demonstrate by means of broad-band spectral modeling that the \gamma -ray flux of the Fermi -LAT source 2FGL J1629.4+8236 may well be accounted for by the high-energy tail of the inverse-Compton continuum of the lobe . Thus , this claimed association of \gamma -rays from the north-west lobe of NGC 6251 , together with the recent Fermi -LAT imaging of the extended lobes of Centaurus A , indicates that particles may be efficiently ( re- ) accelerated up to ultrarelativistic energies within extended radio lobes of nearby radio galaxies in general .