Though there's little canon evidence, the answer is probably yes.
For starters, it's a pretty safe bet that the Borg are aware of the Dominion. We saw in "Endgame" on a map shown by Seven (who probably knows what she's talking about) that the Borg have some kind of facility in the Gamma quadrant. On top of that, the Dominion's cloning technology would be extremely useful to the Borg, so naturally there was probably some investigation on the part of the Borg.
On top of that, the Borg have a species number for the Ferengi, which are an Alpha/Beta quadrant power; which means that the Ferengi are farther away from the Borg than the Dominion is far from the Borg, since the Dominion is in a neighboring quadrant to the Borg. So the Borg probably have at least heard of the Dominion, simply because the Dominion is geographically closer to them than a species they have encountered. Based on this, the Borg have probably encountered the Dominion at least once.
Frequently, during Borg encounters, there's often a survivor or two, such as Guinan, or the people Seven saved in "Dark Frontier", so given that the Borg have probably encountered the Dominion at least once, there was probably at least one survivor, who probably told the Dominion government "Hey, I'll take what I can get in terms of Borg defense, and the Dominion is worth a shot".
So in summary, based on what we know in-universe, it's probable that the Dominion has at least one person who has had some kind of Borg encounter. So from canon sources, the answer is "Probably yes."
Additionally, the Dominion had spies in Starfleet for a while, after first contact, but before the war started formally. Given that the existence of the Borg is public knowledge in the Federation, and given that this is a massive military threat, the Dominion spies almost certainly told their government about the Borg.
On a side note, the Dominion claims to be over 10000 years old "as of the 24th century", whereas we know the Borg were a minor power 900 years ago, having only assimilated a handful of systems and being easy to resist. "Dragon's Teeth". If we extrapolate from that, the Borg are probably about 1500 years old at most. Prior to founding the Dominion, the Changelings were a migratory exploring species. If the Dominion is aware of the Borg, it's definitely from knowledge obtained from member populations, rather than information the Changelings discovered in their peaceful era.