Context. The rotation period of stars is an important parameter together with mass, radius, and effective temperature. It is an essential parameter for any radial velocity monitoring, as stellar activity can mimic the presence of a planet at the stellar rotation period. Several methods exist to measure it, including long sequences of photometric measurements or temporal series of stellar activity indicators. Aims. Here, we use the circular polarization in near-infrared spectral lines for a sample of 43 quiet M dwarfs and compare the measured rotation periods to those obtained with other methods. Methods. From Stokes V spectropolarimetric sequences observed with SPIRou at the Canada-France-Hawaii Telescope and the data processed with the APERO pipeline, we computed the least-squares deconvolution profiles using different masks of atomic stellar lines with known Landé factor appropriate to the effective temperature of the star. We derived the longitudinal magnetic field to examine its possible variation in 50 to 200 observations of each star. To determine the stellar rotation period, we applied a Gaussian process regression, enabling us to determine the rotation period of stars with evolving longitudinal field. Results. We were able to measure a rotation period for 27 of the 43 stars of our sample. The rotation period was previously unknown for 8 of these stars. Our rotation periods agree well with periods found in the literature based on photometry and activity indicators, and we confirm that near-infrared spectropolarimetry is an important tool for measuring rotation periods, even for magnetically quiet stars. Furthermore, we computed the ages for 20 stars of our sample using gyrochronology.