Why would the pickling brine turn cloudy in the jars one month after they were canned?

Sometimes brine does tend to cloud; I'm not sure why. HOWEVER, and this is a big "but" - you need to check the seals on your canning jars. If there's an airbubble, they need to be discarded because they've been heat-processed and contain all kinds of funky toxins and bacteria. In that case, they wouldn't be pickles, they'd be "sick-les." : )