I'm not 100% which version of the Both Queen you're talking about, but for any version the answer to this question is no.
The Queen from The Borg expansion can download a
Event or Interrupt in place of your normal card play. You always only have one of those, so you can only do this once a turn.
The Queen from Shades of Gray let's you download We Are the Borg twice a game, which can be done once or twice on any turn, but to a total of twice a game.
The First Contact Queen lets you download drones in place of card draws, but, in OTF, you are limited to one download a turn so you can only make use of this once.
In theory, you could get three We Are the Borg into play in a single turn by using Shades of Gray Queen to download two copies, plus play one using your normal card play from hand. I'm not sure if there's much point to doing that, because the first function is limited by the download rule, while the second is limited since the second function isn't cumulative.