In this paper, we present Q-U-I JOint Tenerife Experiment (QUIJOTE) 10-20 GHz observations (194 h in total over ≈250 deg2) in intensity and polarisation of G159.6-18.5, one of the most widely studied regions harbouring anomalous microwave emission (AME). By combining with other publicly available intensity data, we achieve the most precise spectrum of the AME measured to date in an individual region, with 13 independent data points between 10 and 50 GHz being dominated by this emission. The four QUIJOTE data points provide the first independent confirmation of the downturn of the AME spectrum at low frequencies, initially unveiled by the COSMOlogical Structures On Medium Angular Scales experiment in this region. Our polarisation maps, which have an angular resolution of ≈1° and a sensitivity of ≈ 25 μK beam-1, are consistent with zero polarisation. We obtain upper limits on the polarisation fraction of Π < 6.3 and <2.8 per cent (95 per cent C.L.), respectively, at 12 and 18 GHz (ΠAME < 10.1 and <3.4 per cent with respect to the residual AME intensity), a frequency range where no AME polarisation observations have been reported to date. The combination of these constraints with those from other experiments confirm that all the magnetic dust models based on single-domain grains, and most of those considering randomly oriented magnetic inclusions, predict higher polarisation levels than is observed towards regions with AME. Also, neither of the two considered models of electric dipole emission seems to be compatible with all the observations together. More stringent constraints of the AME polarisation at 10-40 GHz are necessary to disentangle between different models, to which future QUIJOTE data will contribute.