This is a (super) late answer, but I myself was looking for some information related to gamma-gamma models for monetary value, and came across this. The short answer is yes, the negative values for expected transaction values exposes issues with the underlying dataset used to fit the model.
In case it is helpful for you or others with similar questions, I'll try to illustrate why it's concerning to have $q<1$. The purpose of these spend models is to understand observed spend per transaction with the goal of predicting future spend per transaction at the individual level. The use of a gamma distribution was first proposed by Colombo and Jiang (1999) and was motivated by the observation that if transactions are distributed normal, then 1) it is not bounded below by $0$ for any choice of mean and variance parameters, and 2) you get symmetric spend distributions, when the observed data consistently appears to be right skewed.
Following the paper you refer to, a customer with $x$ transactions values $z_1,\dots,z_x$ is modeled such that $z_i \sim \text{Gamma}(p,\nu),$ and we allow for heterogeneity across customers by also having that $\nu \sim \text{Gamma}(q,\gamma)$. A key observation is that conditional on $p$ and $\nu$, a customer's mean transaction value $\delta$ is $\delta = p/\nu$. Now $\nu$ varies across customers, so you may want to know what the mean transaction value $\delta$ is across all individuals. Denote this random variable $D$. It can be shown that
$$E[D|p,q,\gamma] = \frac{p\gamma}{q-1}$$
which says that the mean transaction value for customers is $\frac{p\gamma}{q-1}$ (showing this is a bit involved, but the way to do it is to derive the distribution and show it is an inverse-gamma distribution with specific parameters and find the expected value given that). In any gamma distribution, the parameters are strictly positive, so $p>0,\gamma >0$, so if you have $q<1$, then it must be that the expected transaction value across individuals is negative.
This should be pause for concern: why is the expected transaction value negative? You can try to validate this by thinking of compensating individuals for each transaction, but this is quite odd and there are other models if this is the kind of situation you are dealing with, and so the fact that your model finds $q<1$ should immediately raise some serious concerns for this reason alone.
As a final point, I think it's nice to better understand
$$
\begin{align}
\mathbb{E}(M\mid p, q, \gamma, m_x, x) & = \frac{(\gamma + m_xx)p}{px+q-1}\\
& = \bigg(\frac{q-1}{px+q-1}\bigg)\frac{\gamma p}{q-1}+\bigg(\frac{px}{px+q-1}\bigg)m_x\\
\end{align}
$$
as noting that it is simply the weighted average of the population mean transaction value $E[D|p,q,\gamma] = \frac{p\gamma}{q-1}$ and the observed average transaction value $m_x = \frac{1}{x}\sum_{i=1}^x z_i$ of a given customer, and the weightings can be fully understood from a bayesian framework as having a prior (the mean average transaction value), and the weight you place on it goes down as you observe more data $x$ on an given individual!
