It's Metro line 4 and it takes 16 minutes. But add on about 5 minutes at each end for getting to and from the metro platform to the train and waiting. Add more if you are unfamiliar with the system and need time to get oriented.
There is no Notre-Dame metro station. The nearest is Cité. It takes about 15 minutes from there to Gare du Nord. The Saint-Michel-Notre-Dame station (metro and RER) is slightly further from both.
The problem here is that Montmartre (the hill itself) doesn't have a metro station. You need to walk down the hill to ANVERS. Then 2 stops on line 2 direction NATION brings you to LA CHAPELLE, from...
follow the signs " RER line E " (there is one direction only as St Lazare is the starting point of the line) and step down at the next station. Here you are.