I was in the same situation couple years ago. I moved from Toronto to Ottawa. I applied my EI in Ottawa. In your case, I'm not sure 'coz it involves two different provinces. You have to call the government to find it. I remember I just provided my ROE, SIN and Driver's Liscence. Again, you have to call them because some of the things might have changed.
As what I know, if you don't have a good reason, you might not be qualified to have EI. If you are sure your reason is good, you only need your ROE to apply for EI anywhere within canada. So it is good you can call Service Canada to know about your reason if it is qualified. We know the rules changed all the time. In the meantime, you could have two weeks parental leave (for father, new in Quebec resident in 2006) which is 75% of your salary at the beginning of your child's born. More info, you and your wife can take turn to have the materity leave which is about one year.
Here's the ROE reasons:
Shortage of Work / End of contract or season
Strike or Lockout
Return to School
Illness or Injury
Quit
Maternity
Retirement
Work Sharing
Apprentice Training
Other
Dismissal
Leave of Absence
Parental
Compassionate Care
Here's the Quit reasons:
Follow Spouse
Return to school
Health school
Retirement
Take another job