Worship meetings of the Religious Society of Friends (or "Quakers") are spent in silence unless people are moved by the Holy Spirit to act. This seems clear enough, however, I do not understand how a worship service comprised singularly of this practice would start or finish. How do congregants know that the worship has finished? How do they know that it is about to/has started?
Each Weekly Meeting (that is, each congregation) has a clerk who is regularly elected by the members of the Meeting to lead all business conducted by the Meeting. Some weekly meetings (such as the School Meeting described by the Friends School of Minnesota) have the clerk of the Meeting or another elder chosen by the Meeting begin the meeting by rising and stating a Query (that is, a question for reflection) which will be the topic of the meeting. Sometimes, on the other hand (as according to this Canadian reference) the Meeting for Worship (the worship service) is considered to begin as soon as even one worshiper enters the meeting room. As Meetings for Worship vary distinctly by Weekly Meeting, there may be other approaches as well.
Typically, the Clerk of the Meeting (or another elder appointed by them) will keep track of the meeting's length, and will stand at the end of the appointed time (45 minutes or an hour in most meetings) and close the meeting (usually with a handshake that then propagates around the room), thanking everyone for coming.
NOTE: Your question, and my answer, relate to the procedure at what are called unprogrammed meetings. Certain branches of the Society of Friends have a very different style of Meeting for Worship called a programmed meeting. These meetings look much like a typical Protestant worship service1, with readings, songs, and a worship leader.
1In my Catholic understanding.