There are 2 ways to handle this in Quicken, at least that I'm aware of. The first is to have Quicken directly link that payment to the account. So in your checking account, it would show as:
payee: Credit card name category: "Credit card in quicken" amount $$
Then in the credit card account, it would show as:
Payee: credit card name category: "checking account" amount $$
(I'm pretty sure - I actually don't like the direct link so have trained it not to do it, and I change it when it does anyway).
The other option is to have categories setup to use in place of the linking categories. I have one called "Credit Card", which all payments on a credit card are assigned to, regardless of the in or out side. This allows my to use the reports without double counting.
So, my checking account reads:
payee: Credit card name category: "Credit Card" amount $$
Then in the credit card account, it reads:
Payee: credit card name category: "Credit Card" amount $$
Since one is a debit and the other is a credit, they net to zero in the reporting, as long as both accounts are included in the report.
I have had problems with a few banks not downloading nicely into Quicken. For those, I export the transaction list from the website in the appropriate format and import it. That works fine.