The short answer to your question is basically no, there's essentially no connection between the prime powers $q^i$ dividing $h_p$ and $h_{-p}$.

It's true that there's a general relationship between the 2-ranks of the class groups of $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{-m})$ as per Franz's answer and the result from Washington you cite (which, incidentally, can be pushed further to 4-ranks, 8-ranks, etc., getting pretty close to a full comparison of the 2-parts of the class numbers given knowledge of the fundamental unit of the real quadratic), but for your case this is almost contentless by genus theory. For other primes $q$, or the class number in whole, it's hard to give a conclusive justification for the "no relationship" claim, though two points bear mentioning:

- Larges tables of these class numbers are available or easily generated, a quick survey of which is pretty compellingly against any correlation; and
- There are all sorts of heuristics out there about how the two classes of class numbers should behave, some of which great imply a lack of relationship. For example, it's very unlikely that $q^i$ dividing $h_{-p}$ could tell you anything about $q^i$ dividing $h_p$ since $h_{-p}$ is non-trivial for all but nine examples, whereas (probably) $h_p=1$ infinitely often.