That dog won't listen to his owner at all. He just keeps barking and barking.
The neighbour's dog won't listen to one word he says. He just won't stop barking.
日本語の通りに訳すと不自然なので、次のように言うと良いでしょう。
ーThat dog won't listen to his owner at all. He just keeps barking and barking.
「あの犬は飼い主のことを全く聞かず、ずっと吠え続けている」
won't listen to his owner at all で「飼い主の言うことを全く聞かない」
to keep barking barking で「ずっと吠え続ける」
ーThe neighbour's dog won't listen to one word he says. He just won't stop barking.
「近所の犬は言うことを全く聞かず、ずっと吠えている」
won't listen to one word he says で「彼の言うことを一言も聞かない」
won't stop barking で「鳴き止まない」
ご参考まで！
The dog that doesn't listen to its owner is barking non-stop.
『全く飼い主の言うことを聞かない犬が吠え続けてる』を英語で表現すると、The dog that doesn't listen to its owner is barking non-stop. となります。
ここで、doesn't listen to its owner は「飼い主の言うことを聞かない」を表し、barking non-stop は「吠え続けている」を意味します。
他の言い方としては次のようなものがあります：
The dog ignores its owner's commands and keeps barking.
The dog won't obey its owner and is continuously barking.
カナダ 
アメリカ合衆国 
