What could cause a battery hedge trimmer to have white smoke come out of it?
Your hedge trimmer sounds like it has died or is in the process of dying.
A quick google search confirms that this can occur with Black &amp; Decker hedge trimmers, although it does not appear to be wide spread. Because you didn't provide the model it isn't possible to give you a more specific reason why it has failed. My opinion is that when you were having difficulty cutting larger stems, the motor was overworked and this is what led to the failure.
Random reviews about white smoke coming from a electric Black &amp; Decker hedge trimmer

unfortunately, mine is broken. I just purchase it from walmart and while using it a smoke started coming out. I took it back to walmart for replacement but they told me to contact black and decker for a replacement.
It seemed to be working great initially and trimmed small forsythia bushes great. However within a short period it started to clatter. I never cut anything larger than about 1/8&quot; and only recharged the batter once. After 10 minutes of use after the second battery charge dense white smoke poured out of the motor compartment, and that was that.

The smoke you are seeing is from the electric motor.

Is it safe to use this trimmer again?

Yes it is safe to continue using the hedge trimmer, however the electric motor will continue to fail until it doesn't work anymore if it is even working now at all.
As noted by Wayfaring Stranger, the electric motor may survive a little over heating, but from what I've read specifically about this issue with B&amp;D electric hedge trimmers is that it's finished or will be very soon.
If it ceases to function, unfortunately you will have to return the product or purchase a new one.