The new generation of broad-band radio continuum surveys will provide large data sets with polarization information. New algorithms need to be developed to extract reliable catalogs of linearly polarized sources that can be used to characterize those sources and produce a dense rotation measure (RM) grid to probe magneto-ionized structures along the line of sight via Faraday rotation. The aim of the paper is to develop a computationally efficient and rigorously defined source-finding algorithm for linearly polarized sources. We used a calibrated data set from the LOw Frequency ARray (LOFAR) at 150MHz centered on the nearby galaxy M51 to search for polarized background sources. With a new imaging software, we re-imaged the field at a resolution of 18''x15'' and cataloged a total of about 3000 continuum sources within 2.5{deg} of the center of M51. We made small Stokes Q and U images centered on each source brighter than 100mJy in total intensity (201 sources) and used RM synthesis to create corresponding Faraday cubes that were analyzed individually. For each source, the noise distribution function was determined from a subset of the measurements at high Faraday depths where no polarization is expected; the peaks in polarized intensity in the Faraday spectrum were identified and the p-value of each source was calculated. Finally, the false discovery rate method was applied to the list of p-values to produce a list of polarized sources and quantify the reliability of the detections. We also analyzed sources fainter than 100mJy but that were reported as polarized in the literature at at least another radio frequency. Of the 201 sources that were searched for polarization, six polarized sources were detected confidently (with a false discovery rate of 5%). This corresponds to a number density of one polarized source per 3.3 square degrees, or 0.3 source per square degree. Increasing the false discovery rate to 50% yields 19 sources. A majority of the sources have a morphology that is indicative of them being double-lobed radio galaxies, and the ones with literature redshift measurements have 0.5<z<1.0. We find that this method is effective in identifying polarized sources, and is well suited for LOFAR observations. In the future, we intend to develop it further and apply it to larger data sets such as the LOFAR Two-meter Survey of the whole northern sky, LOTSS, and the ongoing deep LOFAR observations of the GOODS-North field.