Go back to basics and ask, "What is a correlation?" The right answer is that there are many measures of "correlation" when broadly considered as pairwise metrics for demonstrating association between variables. The naive answer is that it's a Pearson correlation since that's the most commonly taught and known form. But Pearson correlations only measure pairwise, linear association and, moreover, have a rigorous requirement for interval or ratio scaled data. Spearman correlations, on the other hand, measure monotonic association, e.g., between ordinally scaled variables. For financial data which don't always meet the requirements of a Pearson, Spearman correlations are a much more sensible metric.
There are also many, many measures of association for categorically scaled data, as well as nonlinear measures of association such as "distance" correlations, and so on.
In addition, both Pearson and Spearman correlations range between $-1$ and $1$. Given that (and Street vernacular notwithstanding), it's completely erroneous to speak of them in percentage terms.
Not knowing where you got this rule of thumb of a 12 period lag resulting in an "85%" correlation between GDP and PMI, what is your goal? In other words, why are you even interested in replicating such a hoary convention? Moreover, you have such a long time series from Quandl -- back to 1950 -- what is it that makes you so concerned about the difference in the units of time?
What would I do to address your question? I would merge the two series with 3 monthly periods per quarter and do a whole lot of exploratory analytics: scatterplots, time lines, etc. Based on that, next I would run both Pearson and Spearman correlations using different lags, e.g., 1 quarter up to whatever you think a reasonable maximum # of lags is. Then, I would examine where the association was maximized. That's it.
Of course there are more rigorous ways to answer the question that go beyond simple measures of pairwise association, but that's not what you're asking for.