Block not showing up
Your implementation of the block hooks is wrong.  Try the below:

function quick_links_block_info() {
  $blocks['quick_links_staff'] = array(
    'info' =&gt; t('Quick Links'),
    'status' =&gt; 1,
  );
  return $blocks;
}

function quick_links_block_view($delta = '') {
  $block = array();
  switch ($delta) {
    case 'quick_links_staff':
      $block['subject'] = t('Quick Links');
      $block['content'] = quick_links_get_links();
      break;
  }
  return $block;
}


function quick_links_get_links($department = '') {
  return t('Hi');
}


I've namespaced the block, and changed the status from TRUE to 1, as per the hook_block_info() docs.  I've also removed the assignment of $delta, as this is passed through automatically.

REFERENCES

hook_block_info()

hook_block_view()