Okay guys, there are a couple things i wish to rectify here... Also, OP, i've tried to answer your question at the bottom of this post:
Disclaimer: i have bitcoind running on a linux server, i haven't updated for a while, so if something substantially changed in the last couple of versions, this post might be inaccurate. Also, i usually run bitcoind, not bitcoin-qt (the gui). They use the same blockchain, so everything for bitcoind should be more or less the same for bitcoin-qt:
1) it can take days/weeks to synchronise bitcoin core (but defenately not hours, unless you have access to a 64 core XEON server with 256 Gb ram and RAID5 SSD disks on a really fast network). It depends on many factors: your cpu, the speed of your disks, your memory, your internet connection. The last time i synced the blockchain, it took me over 2 days on a server with 16 Gb ram, 15k rpm disks, 4 XEON processors and a really fast network. You can check how far along the synchronisation is by either looking at the bitcoin-qt gui, or looking in the debug.log (since i run bitcoind i only know/can verify to look into the debug.log, but if i remember correctly, the gui also indicated how much of the blockchain was already synced... I did not verify this tough).
2) On my server, i have a dedicated bitcoin-user, the partition in which the blockchain resides is 90+ Gb, and contains allmost only block data. It is allmost full.
OP: before you do ANYTHING, ALWAYS backup your wallet.dat!!! in linux you can find this file in ~/.bitcoin/wallet.dat if you did a default installation, in windows it's somewhere else, you can either use google, or use your windows searchfunction to find it.
OP: to answer your question, a very little lecture... I've tried to keep the explanation newbie-friendly and might have omitted some details, so you cannot use this explanation as a fact, more as a tool to explain what is happening:
A wallet is a piece of software that has a specific task: maintaining your private keys. A private key is the basis of bitcoin. It is (as the name says) a piece of text that you NEED to keep PRIVATE. It is used to controll your bitcoins. Your address is derived from this private key, but can be shared. If somebody gets your private key, he controlls your address and can send your coins to himself!!! I'm tired, and for this little lecture it's a bit to much to give the technical explanation of how this works exactly.
Bitcoin-qt (the client you installed) manages your private keys. It is designed to generate a new private key (and a new address) each time you send coins (to collect your change). It also generates a new private key each time you request coins. Your wallet might manage hundreds of private keys!!! Electrum is a different wallet, it also manages private keys, the big difference is that electrum only download (very small) headers. It uses a central server to get all other information, but keeps your private keys safe on your pc (so the central server cannot get its hands on your keys => it's pretty safe to use electrum).
Now, if you're 100% sure you have only used your wallet to RECEIVE this ONE incoming transaction, and you HAVE NOT used your wallet to send coins, ask more than 1 transaction,... you can try to export your private key for this address.
I could try to explain how to do this, but people have already made this explanation before me:
http://docs.bitshares.org/muse/migration/howto-exporting-wallet-clients.html (look for the explanation of bitcoin qt).
Importing a private key into electrum is pretty easy: just open electrum... If it's the first time you open it, it'll show a wizard, otherwise go to file=>new/restore => restore a wallet or import keys => paste your private key...
If you don't see all your funds in electrum, there must have been more than one address/private key in use in bitcoin-qt, and you can either export these other private keys to, or just wait it out using bitcoin-qt.
I must say i'm rather proud of a new user chosing bitcoin. You made the right decission of chosing a desktop wallet, but you were unlucky because you did this in a hurry and chose a wallet that required a lot of time to synchronise (altough it's a great wallet, it's the basic wallet and it's safe and has a lot of functionality). I hope you stick around the bitcoin community after this first try, it's a good community and a good idear, you just have to give it a chance 
Disclaimer2: it's late in my country, i'm tired and going to bed... It might be wise to wait untill a second longtime member verifies my steps before actually doing them!
