That is an interesting question. There are several verses in the Bible that suggest that they do.
In Numbers 22:28 it says, "And the LORD opened the mouth of the donkey, and she said unto Balaam, What have I done unto you, that you have smitten me these three times?" If God can make a donkey talk, what else does the donkey know?
In Proverbs 12:10 it says, "A righteous man regardeth the life of his beast: but the tender mercies of the wicked are cruel." God has regard for His animals and expects us to be good stewards of them.
"Behold the fowls of the air: for they sow not, neither do they reap, nor gather into barns; yet your heavenly Father feedeth them. Are ye not much better than they?" (Matthew 6:26) God takes care of His animals and some may argue that they instinctively know what to do, but He certainly has the power to communicate with them.
"For we know that the whole creation groans and travails in pain together until now." (Romans 8:22) "The whole creation" refers to animals as well as people.
"For it is written, As I live, says the Lord, every knee shall bow to me, and every tongue shall confess to God." I take this literally to mean that animals will also recognize and bow down to God in some way.
Animals cannot sin and they do not have religion. The Bible was written for humans who God gave the role of caretaker of the earth and His creation. However, just because the Bible doesn't say that animals know God doesn't mean that they do not. They just don't need the salvation message of the Bible.
It was God who called the animals to load the ark, so I tend to believe they know Him at some level. I think God holds His animals close - why else would He have had them come to the ark to be saved in the flood? I think it may be a statement of value as well as a test of obedience for Noah. God could have easily re-created the animals rather than having them load into the ark!
I think God values is creation and has some sort of relationship with his animals, though I do not wholly understand what that is. "Are not two sparrows sold for a farthing? and one of them shall not fall on the ground without your Father." (Matthew 10:29)
We do know that the animals will be in the next life: "The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them." (Isaiah 11:6)
So while I do not think animals have or need "religion," I do think that God watches over them and they know Him at some level - or will.