I bet that you never thought that Clint Eastwood had any connection at all to Star Trek, now did you. Well, I didn't know of any, and I know a lot about both.
In any event, there is a connection, and you're looking at it. Unless this picture of Clint with Leonard Nimoy and William Shatner was taken long after the series run, those are not Clint's children - though the girl might just be Kimber Eastwood, who would have been about four years old during the last season of the series. Kimber wasn't even known about publicly for another twenty years, so this is kind of an odd photo. So, let's tentatively identify this as coming from some kind of open house at Paramount during the 1968-1969 season, when Clint would have just become a superstar due to his Spaghetti Western films.