https://account.sonyentertainmentnetwork.com/liquid/reg/account/forgot-password!input.action
The Link above can reset your PSN Accounts Password
1.. Enter in the email address that is associated with the PSN Account
2.. Enter in your DOB that you have registered to the PSN Account
3.. Make a new Password
4.. Done
How long have you not used the Account??
If Support are saying it don't exist.. then it may have been deleted by Sony.. If an Account is not used for over 24 Months.. they can delete it.. Not sure if they actually do, but it's now in the T&C's that they have that option.
or did you try "Live Chat" only a few Countries have this option to contact Support, if you are coming in from a Country that is not Supported.. then it shows "Account don't Exist".
Your OP is a bit unclear on certain matters.
Ava