Do the canonical writings of any Christian group say if animals in Eden were carnivorous, or when animals on Earth became carnivorous?
Yes, the canonical writings of some (most?) Christian groups does say if animals in Eden were carnivorous and when they became carnivorous; and as for your title (bible as opposed to canonical Christian writings), also yes it is addressed.
That is the blunt answer and technically a full one considering your question. However, you likely want to know not only if it says but also what it says.
Genesis 1:29-30 specifically states that people and animals were given plants for food.
Genesis 9:2-3 (post-flood and therefore also post-Eden) specifically states that animals were given into our hand, that every moving thing that lives can be eaten, just as plants were previously given to us to eat.
For those who use the book of Genesis, which is the vast majority of Christians, it is clear. Humans were told to be herbivores in Eden and until after the flood, and animals were too.
The best argument against this is to say that, although it is clear that humans were told to eat plants and not meat until after the flood, animals were told to eat plants but are left out of the post-flood carnivorous/omnivorous passage. I would counter this with the fact that the carnivorous nature of animals is never mentioned until the same passage that tells humans that they can also eat meat ("I will demand an accounting from every animal."). Because the herbivorous nature of humans and animals is so closely linked in Genesis 1, you could also just take them together as one.
Summarized simple answer: Humans and animals in Eden were all herbivores, and they were not allowed to eat meat until after the flood.
Does this mean that there was no killing or meat-eating before the flood? No. We were herbivores by design, not allowed to eat meat, but since humans betrayed God's design, it is safe to assume that there was probably killing and meat-eating going on. We do know there was murder, as there were two murders recorded in Genesis 4: Cain murdered Abel, and Cain's descendent Lamech murdered a man who injured him. Consumption of meat is not recorded in Genesis during this period.
We now have a rough timeline to answer your question even better. You can separate the history into three periods according to your question. One of no meat-eating, one of meat-eating, and one in between where there was probably meat-eating even though it was not allowed.
The bible is clear on this. The real issue that causes disagreements on this topic between people who understand what the bible says is not that they interpret it differently; the real issue between such people is whether or not you believe that the bible is correct when it states the above-mentioned points. Many people do not believe those points, and that is a separate question, one which Bork and David got into. Though a separate question, it is also a good one.
The short answer is "yes, sort of".
The only sub-group within Christianity that I'm aware of that cares about whether the animals in the Garden of Eden were originally carnivorous or not are the Young Earth Creationists.
To other groups, it's simply a non-issue. Only within the young-earth paradigm do you get the idea that there was no physical death before Adam's sin, and thus it's only from that view that there's a necessity to explain what the animals ate before the fall.
Example of the YEC teaching on the subject can be found here.
Going further, understanding the YEC mindset, the only reason to believe in a young-earth creation as literal history is because of the belief that the Bible, including the first portion of Genesis, is meant to be taken literally.
The reason this is important is that the Bible is considered to be the only authoritative source of Truth, particularly among these groups.
So, to these groups, the only thing that can be considered "canon" is the Bible itself.
While the Bible doesn't use the word "herbivore", it does say this in Genesis 1:29-30:
So, a literal interpretation of verse 30 implies that all living things were given plants to eat. It doesn't exactly say "They can't eat meat", but it does say that God has given plants to eat to every beast of the earth and fowl of the air - and every would include those that are carnivorous today.
Most YEC groups do take this to mean that originally all animals were herbivorous.
With that said, there are dozens of alternate ways that the genesis account is understood. It could be just as easy to say that God gave them plants but that doesn't mean that it's ONLY plants that they were given. It could be that they were able to eat meat, but that just wasn't mentioned. And of course, to anyone other than the YEC adherents, it's a non-issue.
As a YEC - I think David Stratton has presented a good overview of our belief regarding animals (and human beings) being herbivorous before the fall, and that we use the Bible as our only canonical writing (though as I commented on his answer, we do not believe that every word in the Bible is meant to be taken literally, there are many figurative and poetic portions in the Bible)
As to when animals (and humans) became carnivorous, we would point to God's instructions to Noah and his family after the flood:
Genesis 9:1-6 NIV Then God blessed Noah and his sons, saying to them, "Be fruitful and increase in number and fill the earth. (2) The fear and dread of you will fall on all the beasts of the earth, and on all the birds in the sky, on every creature that moves along the ground, and on all the fish in the sea; they are given into your hands. (3) Everything that lives and moves about will be food for you. Just as I gave you the green plants, I now give you everything. (4) "But you must not eat meat that has its lifeblood still in it. (5) And for your lifeblood I will surely demand an accounting. I will demand an accounting from every animal. And from each human being, too, I will demand an accounting for the life of another human being. (6) "Whoever sheds human blood, by humans shall their blood be shed; for in the image of God has God made mankind.
The issue of whether animals and humans were carnivorous from the start is far from a "non-issue." It is central to the understanding of what sin did to the earth, and why Jesus needed to come to earth. When God creates the universe, the earth, animals, and humans, he stops after each day's work and evaluates what he created. When he does so, you read: "And God saw that it was good." This is repeated again and again. If you believe that God created the world with carnivorous animals and humans, then God is looking at a world filled with killing and death and declaring it to be good.
In the Bible there are many passages that talk of Jesus' redemptive work overcoming death. Why would this be the case if death is a good thing?
Romans 5:12 NIV Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned--
Romans 5:18-19 NIV Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people. (19) For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous.
The Bible clearly links death with sin, and says that Jesus has overcome sin and death. Therefore we have the hope of death being removed when Jesus returns, described in the prophecy given to Isaiah:
Isaiah 11:6-9 NIV The wolf will live with the lamb, the leopard will lie down with the goat, the calf and the lion and the yearling together; and a little child will lead them. (7) The cow will feed with the bear, their young will lie down together, and the lion will eat straw like the ox. (8) The infant will play near the cobra's den, the young child will put its hand into the viper's nest. (9) They will neither harm nor destroy on all my holy mountain, for the earth will be filled with the knowledge of the LORD as the waters cover the sea.