Looking at the forum I haven't seen this posted yet, but the card reinforce, can this be used when you there are no reinforcements after or before the overlords turn?

I don't see why not. Even if the quest rules don't provide for any regular reinforcements, this card can still be played (provided you have a monster group on the map that's below it's group limit.)

The card must be played at the start of the Overlord's turn, so not "after."

Per "The Golden Rules" on page 20, the Quest Guide always wins over the Rulebook and the Cards always win over the Quest Guide (and/or Rulebook.) So even if a particular quest had a rule to the effect of "the Overlord may not reinforce monsters during this quest," the "Reinforce" card would still trump that.