Open your owner’s manual. It will show you exactly what maintenance work is needed.
Timing belt replacement is critical. The timing belt does not give any warning before it goes and when it goes the whole engine may go. It also happens without warning so you could find yourself with no power an a semi coming on fast or just slowing to a stop in the middle lane of the freeway.
Has the timing belt been replaced? They generally need to be replaced at about 100,000 miles or X years, which ever comes first. It would appear you are on borrowed time.
[i] Is this maintenance work necessary?[/i] All maintenance work listed in that owner's manual is needed for anyone who does not want to pay the price. With proper maintenance, your Accord should have no problems going another 100,000 miles.
Now about the [i] I got a quote for just under $1000 from a Honda dealer. [/i]
Dealers are no better (or worse) than independent mechanics for almost anything you might need done on your car. They will almost always charge more per hour and often more for parts and supplies. They also tend to look at repairs a little different than the independent.
A dealer may well recommend work that strictly may not be needed, but could be connected to the problem or maybe replace a part when a little repair would fix it ALMOST as good a new.
There is no need to bring your car to the dealer for any service other than service that is going to be paid for by a recall or original warrantee. During the warranty period be sure to have all required (as listed in the owner's manual) maintenance done and to document all maintenance work.
I suggest that most people would be better off finding a good independent (Not working for a chain) mechanic.
Note: Never ever use a quick oil change place. They are fast cheap and very very bad.