Listing custom taxonomy terms by first letter

Add this on function.php

add_filter( 'terms_clauses', 'terms_clauses_47840519', 10, 3 );
function terms_clauses_47840519( $clauses, $taxonomies, $args ){
    global $wpdb;

    if( !isset( $args['__first_letter'] ) ){
        return $clauses;
    }

    $clauses['where'] .= ' AND ' . $wpdb->prepare( "t.name LIKE %s", $wpdb->esc_like( $args['__first_letter'] ) . '%' );

    return $clauses;

}

and call it with something like

$terms = get_terms( array(
    'taxonomy' => $taxonomy,
    '__first_letter' => 'a', // desired first letter
) );

Search custom taxonomy term by name

// We get a list taxonomies on the search box
function get_tax_by_search($search_text){

$args = array(
    'taxonomy'      => array( 'my_tax' ), // taxonomy name
    'orderby'       => 'id', 
    'order'         => 'ASC',
    'hide_empty'    => true,
    'fields'        => 'all',
    'name__like'    => $search_text
); 

$terms = get_terms( $args );

 $count = count($terms);
 if($count > 0){
     echo "<ul>";
     foreach ($terms as $term) {
       echo "<li><a href='".get_term_link( $term )."'>".$term->name."</a></li>";

     }
     echo "</ul>";
 }

}

// sample
get_tax_by_search('Foo');