Is it possible to create a predictive model for a dataset that consists of only positive occurrences of the dependent variable?
My dataset would only contain data about earthquake occurrences and no
data about non-earthquake occurrences as that would basically be any
other period of time which is not kept in the dataset.

It would be nice if you could specify the features used for the prediction.

In that case, I assume a decision tree or logistic regression would
not work as we don't have a dichotomous dependent variable (as the
occurrence only gets into the dataset if an earthquake occurred).

Here you are right!. So it's better to resort to anomaly detection algorithms for the same. Here are some examples

Unsupervised anomaly detection
Outlier detection
Anomaly detection in python