There are a few methods but like most scientific work it isn't always conclusive.
Wind direction. A locally sited meteorological station with wind speed and direction would be your best clue. Otherwise a nearby met station from an airport or site like weather underground might help. More sophisticated would be a back trajectory, which NOAA has a free tool you can use online but it takes some effort to learn:
Similarly upwind and downwind sampling. Setup in multiple locations around a suspected source and see if results correlate with wind direction.
Other pollutants. If you happen to have a complete set of air quality instruments at your site then looking at the other pollutant concentrations can be another clue. Likewise you can speciate the PM itself by using a x-ray scanning microscope. Other test are available like digestions with atomic absorption for metals, elemental carbon (burning to CO2) testing, etc. Then you need to look at what is nearby and what you would expect to see from the various potential sources. for example if you find X it COULD be from Y:
black carbon - burning wood, forest fires, agricultural burning
silica, minerals - natural sand/dirt/rocks, quarries, sandstorms
sulfur - coal, refineries, volcanoes
nitrates - diesel, mobile sources
Pollution source to receptor timing correlations. Timing the samples to be collected based on the suspected source operation schedule is another method. For example, if you know a quarry operates only Mon to Fri 7 am to 3 pm take samples when it's operating and also when it's not operating to measure operations vs. background. This is where real time instruments have a big advantage.