No, the point of enabling TRIM is to not induce further write amplification. The zeroing is deferred until the block that is trimmed is over written. It is released from the indefinite garbage collection that exists in a pure SSD implementation that does not know what blocks are important. Some firmwares answers with nothing when requesting a TRIMed block. Others answer with the content of the block as it is on NAND. Either way, the bits are most likely still on NAND.
I think it's dependent on the GC scheme of the SSD in question. From what I have seen, some drives defer it until they'll be written, some zero at the moment the sectors are TRIMmed and some wait for idle + timeout amount of time before working on these sectors.
I have a couple of external SSDs and USB flash drives start to flash their busy lights after letting them idle for a couple of minutes. These "bursts of activity" generally is proportional to the I/O work they have done before the idle period.
One of my flash drives did that GC/zeroing thing when it's ejected. After ejection and drive disconnect, its busy light flashed for some time depending on the work needs to be done. If you didn't hammer it when you used it, it did nothing, but if you did tons of work, it worked up to a minute or so after being ejected.