Can I set up a new account, separate from my current account(s)

The best way to do it is to just start a new Quicken data file for the family member:

File > New Quicken File…

When you create the new file you can then add this family members checking account to it. You would then have two separate Quicken files that you can use separately. When you open Quicken will be able to select which file to open

While this is a valid option, many users can get confused by this and if you need to add any additional accounts for this family member it will get more difficult to manage. It is your choice of course so whichever way you feel more comfortable with is up to you. I just don't like keeping someone else's accounts in my data file.

The information is kept in data file(s).

You already have one for your information. Just create a second data file for the other family member. You can switch back and forth between the data files using the Recent File list on the File menu.

You really don't want to mix the financial records of folks that file their taxes separately.

As I stated in your duplicate post,

it is a very bad practice to mix the financial records of different taxable entities. What would you do if at a later date needed/decided to hand off that information to someone else with Quicken to keep the records? Are you going to give them a copy of your data file to start with.

Create a second data file, you will not regret it later.

I've done it both ways before, myself.

For me, I found it usually best to set up a new Quicken file for the other person. I found it to be the cleanest and easiest way to manage it, especially when it came to generating/managing reports, budgets, taxes, etc. It also made it very easy to later simply pass the data file along to the other person when that person was old enough to and wanted to manage his own finances.

If this family member is an adult or disabled person then I would definitely recommend creating a new file for that person via File > New Quicken File because the odds are that the complexity of the financial data in the file will be greater than you anticipate and will increase over time. This would increase the risk that their financials and yours might get mixed up a bit so a separate file would prevent that from happening.

Also, if you will be acting as a legal Guardian/Custodian or Attorney-in-Fact (POA) for that person, it is paramount that you keep your finances and their's at arms length from each other. You don't want even the appearance of impropriety taking place because that can land you in legal and IRS trouble. The best way to accomplish that is to set up an entirely new and unique data file for that person.

But if it's just going to be simple tracking of only 1 or 2 accounts, especially if this is for your child, and you won't be wanting to take full advantage of the all reports and tools in Quicken, then simply doing Add Account for those few accounts and and holding them as Separate would probably work well, too.

@QuickUserPSP Obviously, we will have to agree to disagree. Mixing tax entities in the same file, regardless of the number of accounts is a bad practice.