Hide post if matches current month and year

Initialize an array before the first loop:

$exclude = array();

Then add the ID of each post within the first loop to that array:

while ($query->have_posts()) {
    $query->the_post();
    $exclude[] = get_the_ID();
    // etc...

Then use that array to exclude those IDs via post__not_in in the second query:

$args = array(
    'post_type' => 'page',
    'post__not_in' => $exclude,
    // etc...