The short answer is no. Babies don't go quiet, or stop moving, before labour. Babies move throughout pregnancy, up to and even during labour.

Do babies move a lot right before birth?

If this is your first baby, they will usually may move down into your pelvis before the birth. This is called 'engaging' and when it happens any breathlessness you've been feeling will probably ease. Second or later babies may also do this, but it may not happen as early.

Does baby move more closer to due date?

The number of movements you feel each day will plateau around this time, but they should not drop off. Your baby should continue to move to his usual pattern as you near your due date. He will even keep moving during your labour. You may notice that his movements feel different in late pregnancy.