Found a solution.
I found a function that answered my question.
Here is the function:
function get_category_by_letter($letter) {
$args=array(
'orderby' => 'name',
'order' => 'ASC',
'hide_empty' => 0);
$categories=get_categories($args);
foreach($categories as $category) {
$catname = $category->name;
$first_letter = substr(strip_tags($catname), 0 , 1);
if(strcasecmp($first_letter,$letter) != 0) continue;
else {
$cats[] = $category->term_id;
$cats[] = $category->name;
}
}
return $cats;
}