Cross sections of the $^{91}\mathrm{Zr}(n,\ensuremath{\alpha})^{88}\mathrm{Sr}$ reaction were measured at four incident neutron energies in the range of 3.9 to 5.3 MeV. The experiment was carried out on the Van de Graaff accelerator EG-5 at the Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Russia. Fast neutrons were produced via the $^{2}\mathrm{H}(d,n)^{3}\mathrm{He}$ reaction with a deuterium gas target. A twin gridded ionization chamber (GIC) was used as the charged particle detector, in which $^{91}\mathrm{Zr}$ back-to-back samples on a tantalum backing were placed at the common cathode. The relative and absolute neutron fluxes were measured using two $^{238}\mathrm{U}_{3}{\mathrm{O}}_{8}$ samples inside the GIC. Cross-section data obtained in this measurement were compared with those in existing data library and from theoretical calculations using talys code. The present data for the $^{91}\mathrm{Zr}(n,\ensuremath{\alpha})^{88}\mathrm{Sr}$ reaction should be helpful in clarifying the discrepancies in various published nuclear evaluation data. An alpha clustering factor is calculated using the statistical model and knock-on mechanism.