After thinking about it for a bit, the relationship is as follows for a rational $q$,
$$K(\text{height}\,q) \leq^+ K(q) \leq^+ 2\,\text{height}\,q.$$
This isn't really a surprising or useful relationship, but I spell out the details nonetheless.
Upper Bound
Recall that the Kolmogorov complexity of a nonnegative rational $q$ is defined as $K(q):= K(k,l)$ where $q=k/l$ in lowest terms. Also recall that all notions of equality and inequality are up to an additive constant, which I will denote as $=^+$ and $\leq^+$. By (the stronger version of) symmetry of information we have
$$K(q)= K(k,l)=^+ K(k \mid l,K(l)) + K(l).$$ Then we have the bounds (where $\log_2$ and $\text{height}$ are rounded down to an integer),
$$K(q) \leq^+ \log_2 k + \log_2 l \leq 2\,\text{height}\,q.$$
$$ $$
I believe this upper bound $K(q) \leq^+ 2\,\text{height}\,q$ should be tight in the following sense. Let $(X,Y)$ be a Martin-Löf random pair in $\{0,1\}^\infty$. Recall, $X$ is a Martin-Löf random if, there is some $C$ such that for all $n$,
$K(X\upharpoonright n) \geq n - C$ where $K\upharpoonright n$ denotes the first $n$ bits of $X$. A Martin-Löf random pair is (by van Lambalgen's theorem) is a pair where $X$ is Martin-Löf random and $Y$ is Martin-Löf random relative to $X$. Then, I believe that this implies that there is a constant $C$ such that for all $n$,
$$K(X \upharpoonright n, Y\upharpoonright n) \geq 2n -C$$
(although, you should check this detail). Then let $q_n$ be the rational gotten by taking ratio of $X\upharpoonright n$ and $Y \upharpoonright n$ (converted via binary into integers). Then we have that there is some $C$ such that for all $n$,
$$K(q_n) \geq 2\,\text{height}\,q_n - C.$$
(I guess I am also using that these $q_n$ are written in lowest terms---that is $X\upharpoonright n$ and $Y \upharpoonright n$ are relatively prime---for sufficiently large $n$. This seems reasonable, but I guess that is another detail that needs to be checked. If it is not true, I think "for all $n$" can be replaced with "infinitely many $n$".)
Lower bound
Since the height is computable from $q$, we have
$$K(\text{height}\,q) \leq^+ K(q).$$
This lower bound is also tight in the following sense. Consider a computable real number like $\pi=3.14...$. Let $\pi_n$ denote the dyadic rational approximation of $\pi$ up to $n$ decimal places. Then there is some $C$ such that for all $n$,
$$K(\pi_n) \leq K(n) + C.$$
For all the $n$ such that $\text{height}\,(\pi_n)=n$---which happens when $\pi_n$ has a $1$ for the $n$th decimal place---we have
$$K(\pi_n) \leq K(\text{height}\, \pi_n) + C.$$
