The FQ is a 20 minute walk at minimum and that's if all you want to do is go to the nastiness known as Bourbon St.
The walk is closer to 40 minutes if you want to check out the Riverfront and French Market area which, imo, is much nicer and a lot more interesting.
NOLA is one of my favorite places and I'm a frequent visitor - usually as a single female.
I'm also not a chicken or a sissy in the least when it comes to urban areas but I find the station itself unpleasant and not well supervised and the area surrounding it desolate and not visitor friendly.
Use Google Maps street view and virtually walk it and see what you think.
My personal suggestion, and the only way I ever go, is to take the Trolley or a taxi - others here may think differently but I like to err on the side of caution.
Also, the Trolley is awesome and a great way to get around.
If you're ever in the FQ for any length of time, you really need to try a pedicab too - I love them and use them whenever possible.
I don't know how much time you actually have before catching the CONO but there is now also a HOHO (Hop On Hop Off) bus that will take you all over the city.
The tours are from 9:30 to 5:30, 7 days a week - there are 11 stops and there's a bus every 30 minutes - tix are $29 for a full day of fun - the whole loop only takes 2 hours if you just want to do that.
The closest stop to the station is at the Hyatt about a 5 minute walk from the station on Loyola.
Check it out here:
Hop On Hop Off New Orleans