Suppose, I've installed a new coin wallet, got synchronized with the network, now I've sent some coins to my wallet. Now, if I close my wallet (and stay offline), would it get maturity after X amount of time (X is the coin maturity time) or would I have to keep my wallet online to get maturity ?
No, your wallet does not have to be online. Maturity depends on the block number in which a transaction was included into the blockchain.
Receiving bitcoins is a completely passive process: The information of the received transaction gets added to the blockchain enabling you to reference it in the future when you want to spend the balance.
The recipient doesn't have to acknowledge, confirm or even notice the transaction in order for it to be settled.