Can I close Bitcoin-QT client and power off my computer just after sending coins to someone or do I need to wait until 1, 2...6 confirmations?
Since there is nothing that shows you that peers have received and are relaying your transaction you might want to leave your client open until there is one confirmation.
In addition to the initial broadcast to peers, there are subsequent re-broadcasts performed by your client if the transaction has not received a confirmation. This re-broadcast generally will happen in under an hour and will continue indefinitely at about that pace until the transaction gets a confirmation.
Without leaving your client open, these re-broadcasts will not occur.
If other nodes have just started up (i.e., a clean memory pool) or have flushed your transaction due to it not having gotten a confirmation after an extended period of time, only the re-broadcast will get the transaction back into that node's memory pool.
Once a transaction has a confirmation though, there is no further need to keep your client open. There are some corner cases where that's not true (e.g., a block reorg occurs and that transaction confirmation no longer is valid, etc.) but in almost every instance once a transaction has a confirmation it then no longer needs to be re-broadcast.
You can just close the client. Once the transaction is sent into the network, all clients received it and miners will include in one of their next blocks. (Sometimes they do not, but that has nothing to do with whether your client is open or not.)
Your client is just a gateway to the peer-to-peer network, meaning that all you need to send/access your wallet is your wallet information (the wallet.dat file when you use the Bitcoin-Qt client). When you backup that information and lose your client, you can always recover your coins on another machine.