button up

1.  Close securely, fasten, as in The house was all buttoned up, or Button up your coat--it's very cold. [Late 1500s]
2.  Also, button one's lip. Hold one's tongue, keep quiet. For example, Please button your lip about the surprise. A variant of this usage, button one's mouth, dates from the 17th century. [Mid-1800s]
3.  Finish successfully, as in I've got this report all buttoned up. [c. 1940]

The verb has one meaning:

Meaning #1: refuse to talk or stop talking; fall silent
  Synonyms: close up, clam up, dummy up, shut up, belt up, be quiet, keep mum