The NANOGrav Collaboration has recently reported evidence for the existence of a stochastic gravitational wave background in the 1–100 nHz frequency range. We argue that such a background could have been produced by magnetohydrodynamic (MHD) turbulence at the QCD scale. From the NANOGrav measurement, one can infer the magnetic field parameters: a comoving field strength close to microGauss and a correlation length close to 10% of the Hubble radius at the QCD phase transition epoch. We point out that the turbulent decay of a nonhelical magnetic field with such parameters leads to a magnetic field at the recombination epoch, which would be sufficiently strong to provide a solution to the Hubble tension problem, as recently proposed. We also show that the MHD turbulence interpretation of the NANOGrav signal can be tested via measurements of the relic magnetic field in the voids of the large scale structure, with gamma-ray telescopes like CTA.