Hi and welcome. 
At the moment, the only way to override that piece is modify the template.
A "quick&dirty" implementation could be to replace, in the file BoardIndex.template.php the code:
// People's birthdays. Like mine. And yours, I guess. Kidding.
if (!empty($context['calendar_birthdays']))
{
echo '
<p class="inline">
<span class="birthday">', $context['calendar_only_today'] ? $txt['birthdays'] : $txt['birthdays_upcoming'], '</span>';
// Each member in calendar_birthdays has: id, name (person), age (if they have one set?), is_last. (last in list?), and is_today (birthday is today?)
foreach ($context['calendar_birthdays'] as $member)
echo '
<a href="', $scripturl, '?action=profile;u=', $member['id'], '">', $member['is_today'] ? '<strong class="fix_rtl_names">' : '', $member['name'], $member['is_today'] ? '</strong>' : '', isset($member['age']) ? ' (' . $member['age'] . ')' : '', '</a>', $member['is_last'] ? '' : ', ';
echo '
</p>';
}
with:
// People's birthdays. Like mine. And yours, I guess. Kidding.
if (!empty($context['calendar_birthdays']))
{
echo '
<p class="inline">
<span class="birthday b-today">', $txt['birthdays'], '</span>';
// Each member in calendar_birthdays has: id, name (person), age (if they have one set?), is_last. (last in list?), and is_today (birthday is today?)
foreach ($context['calendar_birthdays'] as $member)
{
if ($member['is_today'])
{
echo '
<a href="', $scripturl, '?action=profile;u=', $member['id'], '">', $member['is_today'] ? '<strong>' : '', $member['name'], $member['is_today'] ? '</strong>' : '', isset($member['age']) ? ' (' . $member['age'] . ')' : '', '</a>', $member['is_last'] ? '' : ', ';
}
}
echo '
</p>
<p class="inline">
<span class="birthday b-not-today">', $context['calendar_only_today'] ? $txt['birthdays'] : $txt['birthdays_upcoming'], '</span>';
// Each member in calendar_birthdays has: id, name (person), age (if they have one set?), is_last. (last in list?), and is_today (birthday is today?)
foreach ($context['calendar_birthdays'] as $member)
{
if ($member['is_today'] === false)
{
echo '
<a href="', $scripturl, '?action=profile;u=', $member['id'], '">', $member['is_today'] ? '<strong>' : '', $member['name'], $member['is_today'] ? '</strong>' : '', isset($member['age']) ? ' (' . $member['age'] . ')' : '', '</a>', $member['is_last'] ? '' : ', ';
}
}
echo '
</p>';
}
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