He makes me lie down in green pastures
The words "sleeping dogs" immediately call to mind the expression "let sleeping dogs lie." No doubt Klingons know the wisdom expressed in this proverb: don't stir up trouble when you don't have to.
Klingons do, after all, know about pets (the Klingon word for pet is Saj), though their 'dog' is the targh, a fairly fearsome creature; definitely not something to rile unnecessarily.
Let's consider "sleeping dogs" (the undisturbed kind) as an illustration of today's Klingon word: Qot, 'to lie.'
David in Psalm 23 considers God's care for him saying that God:
Qot bIng Daq SuD tI yotlh
The English word 'lie' occurs over 100 times in the World English Bible translation of the Hebrew scriptures, mostly referring to an action like or involving reclining (as opposed to telling a falsehood). The Hebrew text of this psalm uses a specific verb, rabats, that is only used around 30 times. The notion in this word is that of a recumbent animal. This image would be familiar to a shepherd like David. Hovever, I'm not a shepherd, so I find the image of a recumbent animal that comes to my mind is that of a sleeping dog.
The utter peace of my own dog is something I find delightful. Once comfortable, he relaxes so completely that it compels me to settle down beside him as well. That is the sort of peace, of rest, that David is telling us he finds because he knows that the Good Shepherd is leading him.
And it's a peace offered to us as well. Just as God promised the Hebrews in the book of Exodus:
God said, "My presence will go with you, and I will give you rest."
Are you looking to Qot Bing Daq roj, to lie down in peace? Then listen to the Good Shepherd:
Come to me, all you who labor and are heavily burdened,
and I will give you rest.