I love this question. It's one I struggle with myself. 'Cause the most realistic answers probably make boring fiction.
The priorities and values of such empires may be hard for us to comprehend, to the point where, even if they're all ancestrally human, they seem alien to Us now. Technology might be the greatest determiner - the more advanced, the further removed they may be from our notion of human. It's a hard nut to crack - how does one sympathetically portray a character/empire/species when their entire existence is in a matryoshka matrix around a red dwarf, or they're genetically engineered to live in space and communicate via organs we don't have and reproduce through spores or not at all?
Assuming these empires are initially content with the space they occupy (which may not be the case; their goals might be to control an entire supercluster, all the better to manage resources to survive a few trillion years), the borders of these territories may be in flux simply because not all stars roam at the same rates. An occupied system might find itself closing in on another territory - in which case, whose will it be? And stars nova; empires that can accurately predict supernova events may need to bug out of valued territory, and the closest safe space may not belong to them. And not all parts of the galaxy are equal; densities differ regionally, meaning some areas have more raw resources, some fewer, and some systems may be more or less desirable based on proximity, star types/ages, etc. In our specific case, Andromeda is coming at us - do certain models suggest parts of the Milky Way are safer than others? If these factors aren't considered problematic because these civilizations have technology to counter or control cosmological events of these scales, who knows what they'd value? Or maybe there really is only room for one with that sort of muscle?
As for trade, information would be a prime currency. As you point out, resources are plentiful. Sure, planet- or culture-specific products might seem valuable at first glance (wine, sculpture, etc.), but these can be "printed" locally from plans sent at light speed (or via "subspace" or such, if you wanna go that route.) The Trekkian "this replicated banana doesn't compare to the real thing" is nonsense; if you're replicating at a subatomic level (and you'd better be, for food), it's perfect to our taste buds every time (and the printer won't release it unless it is). Owning a DaVinci might still mean something, but when it can be duplicated to its finest structure, will it really matter if it's the original? Could one tell? Technology, literature, poetry, music, philosophy, etc. might be the only things worth trading - and at light speed, they're the cheapest and fastest.
So, say a smallish empire (a few dozen systems around its origin point) is about to lose one of it core stars to a nova. This nova will not only eradicate biospheres on its two inhabited planets, it will also destroy or scatter most of the resources of that system, and its radiation will endanger the nearest systems. They have a couple trillion citizens to move, but the empires on either side won't accept them - is war the solution?
Or one empire has developed a new technology (or discovered an ancient one!) it hasn't shared - black hole farming, energy-free FTL, an entropy reverser, something far-reaching or magical. But the effects of that technology have been detected, and they worry the neighbors. In this future information economy, is withholding tech considered an act of war?
Or one empire has simply up and disappeared (into a universe of their own creation, one with modified laws of entropy that will allow them to prosper for quintillions of years before needing to create yet another?), and for the first time in a thousand years there is "free space" to occupy - now it's a free-for-all to get it, along with whatever amazing infrastructure that other group left behind.
Or, as was pointed out above, one of the empires is collapsing from internal strife. Will sides be taken, spurring inter-Empire conflict?
If nothing else, historical memory can get things moving. The first time empires A and B meet, things go badly. Though peace is achieved, the enmity remains, straining later relationships with others they meet who may seem too cozy with their disliked neighbors. (A tells C not to trust B, B tells D not to trust A, etc.) Bake some of these biases into the various treaties. Once all the players are in place and they realize their various arrangements aren't equitable, they may be looking for excuses to "expand their interests."
