Le date MongoDB funzionano insieme a java.util.Date. Cambia la tua query in qualcosa del genere:
Date start = new java.util.Date(2012, 06, 20, 10, 05);
Date end = new java.util.Date(2012, 06, 20, 10, 30);
BasicDBObject query = new BasicDBObject("Date",
new BasicDBObject("$gt", start)).
append("$lte", end) ));