The (Legends) Star Wars Galaxies features multiple polities which span or exist in more than one Galaxy, though I wouldn't classify most of them as intergalactic civilizations in anything but the loosest terms...

The Yuzzhan Vong were a civilization that leapt galaxies, from their original one to the Star Wars main galaxy.

The Intergalactic Banking clans had branches in the minor, orbiting galaxies that surround the star wars galaxy.

The Silentium and Abominor were apparently from the same galaxy, although I didn't know that until I looked this up because I couldn't remember their names.

Alright, I'm out of Star Wars.

Star Trek? I don't remember any.

Stargate? The Ancients, The Asgard, even Earth during Atlantis, although technically human civilization was intergalactic the entire time. They all speak modern english, after all.

Farscape has a few (or maybe just one?), it seems they exist in an intergalactic civilization (in the same way we live in a planetary civilization), in that there may be multiple cultures and governments but there's plenty of interaction.

Doctor Who seems to have civilization on an intergalactic scale as well.

In that respect, maybe we should just try to figure out which universes have civilization on an intergalactic scale, and which don't.

In that regard, Star Wars definitely does not fit, even with the Intergalactic Banking Clans, because the orbiting dwarf galaxies don't really represent sufficient quality/quantity, but farscape does.