High pollen counts implies green space, which means lower population or area where funds are available to maintain green spaces, which means the is probably just another proxy for income levels (regardless of any attempts to control for income, etc)
The first is that pollen count isn't fully dictated by green space. Different species of plants give off varying amounts of pollen, not to mention the influence of wind patterns on pollen distribution especially over long distances.
The second is that low pop/greener areas in the US also tends to mean rural. Rural is usually connected with lower income, not higher. And it's usually lower income areas which see a greater percentage of reported crimes.
The third is that your claims are not really related to the article. The research was examining crime rates on high pollen days across cities, which would seem to cover both low and high income areas depending on how extensive their coverage was.