Either there is or not. That's a matter of faith. God spoke the word or life was created by an exploding star. There's not 100% either way. So, it's a matter of faith to believe in God or Nature.

Does God have a conscience?

Yes there is.
Genesis 1:27 tells us that humans are created in the image of God. This does not mean that we are similar to God outwardly - God is spirit, and we are flesh. We are created in the image of God in the sense that we are endowed with the qualities that He possesses. Including moral sense and the voice of conscience.
If God did not have a conscience, then people would not have it either.

But the conscience of spiritual persons and people works in different ways.

A person's conscience protects, warns, blames or
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justifies it (depending on the act).
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God is perfect and has a perfect understanding of everything. Therefore, He cannot do wrong (Psalm 19: 9).
Therefore, God's conscience evaluates everything he does.
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Genesis 1 talks about God's creatures. After each created creation, He said "it's good." Those. God evaluated everything that he created.
Gen 1:31 - God looked at all of his creations and saw "that it is all very good."

Conscience for a person is a deterrent.
Conscience for God is an evaluating factor.