PHPでメッシュコードの中心点を取るコードが以下です。
呼び出すのは coordinateFromMeshCode($mesh_code) で引数にメッシュコードを入れればOKです。
4分の1などを出したい場合は末尾の数字を削っていけばOKです。
その際中心出す距離も適宜変更が必要です。

function coordinateFromMeshCode($mesh_code) {
    $first_coordinate = $this->first($mesh_code);
    $second_coordinate = $this->second($mesh_code);
    $third_coordinate = $this->third($mesh_code);
    $forth_coordinate = $this->forth($mesh_code);
    $fifth_coordinate = $this->fifth($mesh_code);
    $sixth_coordinate = $this->sixth($mesh_code);
    $lat = ($first_coordinate[1] + $second_coordinate[1] + $third_coordinate[1] + $forth_coordinate[1] + $fifth_coordinate[1] + $sixth_coordinate[1]) / 3600;
    $lon = ($first_coordinate[0] + $second_coordinate[0] + $third_coordinate[0] + $forth_coordinate[0] + $fifth_coordinate[0] + $sixth_coordinate[0]) / 3600;

    $center_lat = $lat + 3.75 / 3600 / 2;
    $center_lon = $lon + 5.625 / 3600 / 2;
    return [$center_lat, $center_lon];
}

private function first($mesh_code) {
    $lat = substr($mesh_code, 0, 2);
    $lon = substr($mesh_code, 2, 2);
    return [(intval($lon) + 100) * 3600, $lat / 1.5 * 3600]; // 3600をかけて秒に直す
}
  
private function second($mesh_code) {
    $lat = substr($mesh_code, 4, 1);
    $lon = substr($mesh_code, 5, 1);
    return [$lon * 7.5 * 60, $lat * 5 * 60]; // 60をかけて秒に直す
}

private function third($mesh_code) {
    $lat = substr($mesh_code, 6, 1);
    $lon = substr($mesh_code, 7, 1);
    return [$lon * 45, $lat * 30];
}

private function forth($mesh_code) {
    $forth = substr($mesh_code, 8, 1);
    if ($this->is_south($forth)) {
        $lat = 0;
    } else {
        $lat = 15;
    }

    if ($this->is_west($forth)) {
        $lon = 0;
    } else {
        $lon = 22.5;
    }
    
    return [$lon, $lat];
}

private function fifth($mesh_code) {
    $fifth = substr($mesh_code, 9, 1);
    if ($this->is_south($fifth)) {
        $lat = 0;
    } else {
        $lat = 7.5;
    }
    
    if ($this->is_west($fifth)) {
        $lon = 0;
    } else {
        $lon = 11.25;
    }
    
    return [$lon, $lat];
}

private function sixth($mesh_code) {
    $sixth = substr($mesh_code, 10, 1);
    if ($this->is_south($sixth)) {
        $lat = 0;
    } else {
        $lat = 3.75;
    }

    if ($this->is_west($sixth)) {
        $lon = 0;
    } else {
        $lon = 5.625;
    }
    
    return [$lon, $lat];
}

private function is_south($code) {
    if ($code == 1 || $code == 2) {
        return true;
    }
    return false;
}

private function is_west($code) {
    if ($code == 1 || $code == 3) {
        return true;
    }
    return false;
}