Will we be perfect in Heaven?

Ecclesiastes 12:7 NIV

and the dust returns to the ground it came from, and the spirit returns to God who gave it.

I take that to imply that our physical bodies do not go with us. However it is a gnostic idea. It is custom of modern Christianity to believe that we return to our bodies. Either way, our bodies are not perfect.

We are who we are, in Heaven and on Earth - with free will and all. I suspect that I will not play the harp well the first time in Heaven :)

Another little bit about the distinction between our Earthly bodies:

1 Corinthians 15:42-44 NIV

So will it be with the resurrection of the dead. The body that is sown is perishable, it is raised imperishable; it is sown in dishonor, it is raised in glory; it is sown in weakness, it is raised in power; it is sown a natural body, it is raised a spiritual body.

If there is a natural body, there is also a spiritual body.

We will receive a glorified body like that which was shown to the Apostles on Mt Tabor at the Transfiguration.

The Transfiguration gives us a foretaste of Christ's glorious coming, when he "will change our lowly body to be like his glorious body." But it also recalls that "it is through many persecutions that we must enter the kingdom of God":

CCC 556

And that Catechism entry specifically references the following scripture:

Philippians 3:20-21 (DRA)
²⁰ But our conversation is in heaven; from whence also we look for the Saviour, our Lord Jesus Christ,
    ²¹ Who will reform the body of our lowness, made like to the body of his glory, according to the operation whereby also he is able to subdue all things unto himself.
   

(at the resurrection, not right after death)

Scripture doesn't address accidents and the like. In the context of Christianity, this means the only valid answer is "We don't know".

The King James and other versions do use the word "perfect" in several places in the Bible when describing saved state:

2 Timothy 3:17 That the man of God may be perfect, thoroughly furnished unto all good works.

Hebrews 6:1 Therefore leaving the principles of the doctrine of Christ, let us go on unto perfection; not laying again the foundation of repentance from dead works, and of faith toward God,

Hebrews 10:14 For by one offering he hath perfected for ever them that are sanctified.

James 2:22 Seest thou how faith wrought with his works, and by works was faith made perfect?

1 Peter 5:10 But the God of all grace, who hath called us unto his eternal glory by Christ Jesus, after that ye have suffered a while, make you perfect, stablish, strengthen, settle you

The above are just the first few I found, but there are plenty more. However, the meaning in the original language is usually better translated as "Mature", "justified", or "sinless" depending on the context and the original word(s) used. They don't indicate the lack of accidents in the manner you described.

Only God is perfect, and (except for Mormon teaching) we will never be God. Ergo, we will never be perfect.

Going a bit deeper into the word, however, when the Bible uses the word perfect, it is typically the Greek word telos. Telos does not imply a state of being, but rather a direction toward the end for which a thing is made. It has the sense of a target, minus the actual arrival.

Note: This doesn't mean our bodies won't be a heck of a lot better - our new bodies will be glorified. That which was sown perishable is raised unperishable. Our bodies will not be affected by sin, and that makes them way better than we can imagine now.

But perfect is a different level entirely.

Think of it as two ants and looking at my belly and saying "Its the largest thing ever!" No, it's pretty massive, but there are larger things. (Until I eat them)