Watch how a string quartet hits the first note when they play. The first violin makes eye contact with the other three and gives a small nod. Similar to how they end the piece, which often has some ritardando and dramatic change in dynamics.
I have played two-handed piano with a friend and I give the nod to start. It is less clumsy than counting out loud (which also works).
An interesting 2012 movie about a string quartet (starring Philip Seymour Hoffman and worth seeing) called A Late Quartet
shows a number of vignettes about how they launch themselves.