NO!
I was looking to kill some of it today. There used to be a weed and feed fertilizer that you spread with dew still heavy, and the theory was that it would dry on the leaves of the clover and broad weeds, and it would kill it. No need to water it in.
I got up this AM looking for it at Lowe's and they don't have it anymore. I used to spread this stuff just for this reason this time of year.
Do NOT broadcast or spray round up even if the St.Augustine is dormant. The grass may be dormant, but the Roundup will still find its way to the roots.