Decays of B(s)(0) and B(0) mesons into J/ψ π+π-π+π- final states, produced in pp collisions at the LHC, are investigated using data corresponding to an integrated luminosity of 3 fb-1 collected with the LHCb detector. B(s)(0) → J/ψ f1(1285) decays are seen for the first time, and the branching fractions are measured. Using these rates, the f1(1285) mixing angle between strange and nonstrange components of its wave function in the qq structure model is determined to be ±(24.0-2.6-0.8+3.1+0.6)°. Implications on the possible tetraquark nature of the f1(1285) are discussed.