The question is rather hard to answer as the measure of intelligence can be defined within many different aspects. I am not aware of specific 'breeding' approaches in this context.
Moreover, it also differs depending on the animal models. What I basically want to say is that intelligence is an emergent phenomenon from vastly complex molecular and environmental factors.
One example of a 'measure' is the **cogntivie factor (g)** both in humans and animals.
> Could it be possible to do so using gene modification?
Nevertheless, there are genotypes related to cognitive function. Famously **FOXP2** was coined the "language gene". (Spoiler: it's not that simple). In humans, **this article** summarizes such aspects from a genetic basis.
To give another example – as I commented before – researchers looked at **transgenic monkeys** to study brain development. In this study, the transgenic monkeys showed better short-term memory and shorter reaction time compared to the wildtype.