From Paris Montparnasse station to the Gare Du Nord by Metro

A service of underground commuter trains, known as RER trains, can speed travellers from the Gare De Lyon to the Gare Du Nord in less than 10 minutes, but there is no such service between the Gare Montparnasse and the Gare Nu Nord on the other side of the city centre.

So if you don’t want to take a taxi, the taking the Metro between these two stations is the quickest public transport option.
It looks straightforward, as Metro Line 4 provides a direct link between Montparnasse and Gare Du Nord (direction Porte de Clignancort), but it is not step free, so isn't suitable for those with restricted mobility, or if you want to avoid having to carry luggage.

Taking the Metro is also at least 5 x cheaper than taking a taxi and almost certainly faster too, despite it being a 14 stop journey; so you will be on the train for around 20 - 25 minutes.

Taking the bus
Though if time is your side bus line/route 91 is now the core inter-station bus service in central Paris.
It heads to the Gare Du Nord from this stop accessible via Hall 2; which exits on to the Place de Cinq Martyrs du Lycée Buffon
And from this stop

If you take the metro, line 4 is also the 2nd busiest Metro line 4 in Paris; so try to time it so that you’ll be using outside of rush hours.
The typical total transfer time, from stepping off a TGV at Gare Montparnasse to being at the Gare Du Nord will be around 35 - 40 minutes.

However, if you are connecting to the Eurostar and have Standard or Standard Premiere tickets, you will need to allow a minimum of another 30 mins to pass through check-in etc.
So when booking connections into Eurostar trains avoid any journey with less than 1hr 20 mins between the scheduled arrival at Montparnasse and the departure time from Gare Du Nord

Walk ahead to the front of the train and exit on to the main concourse (the only exit from the platform/voies).

Many TGV trains that arrive at Montparnasse can be 20 coaches long, so if your reserved seat is towards the rear of such a train, it will take more than 5 mins before you step on to the main concourse at the station.

If time is tight move towards the front of the train before it arrives in the station.

When you have reached the Hall 1 concourse (where the main departure board will be hanging from the roof), keep going ahead of you until you are on the top level of the mall that's housed within the station.
You will see flights and stairs, both of which connect this Hall 1 to the entrance to the Metro station, which is located two levels below.

Though a tad annoyingly both escalators can be switched so that they don’t allow people to descend.
When that is the case and you have luggage, it can be worth seeking out the somewhat hidden lifts/ascenseurs.

The Metro station is located at the foot of an atrium across the front of the station which houses these stairs and escalators.
There are multiple sets of these esclators, but they are flights on either end of this atrium on the far sides of the building.
The entrance to the metro station is in the middle of this atrium, so you may have to seek it out - it has a raised circular area immediately in front of it.

There are also now slopes/ ramps on this level, which give step-free access to the Metro entrance.

Your end-2-end train journey ticket will not include the Metro, so you will need to buy the Metro-Train-RER ticket, which costs €2.50.

However Paris has pretty much transitioned to a Pay As You Go system named Navigo for its public transport network.
The RER ticket(s) for the journey on to the Gare Du Nord have to be loaded on to a physical / plastic Navigo Card, or purchased on either one of two apps - the Île-de-France Mobilités app, or the Bonjour RATP app.
One card per person, or one app per person.

If you opt to use a card you can tap it on the ticket gates, but if you opt to use an app, you tap the mobile device on which the app is installed, on the ticket gates.

If you don't want to use an app (see below), a Navigo Easy card, which costs €2 is the best option for a journey from Montparnasse to the Gare Du Nord, because single journeys can be loaded on to it.
Though if you will be making a return journey which also involves a cross Paris transfer, you can buy two tickets / journeys and load them on to the card.

Some of the ticket machines in the Metro at Montparnasse will sell the Navigo Easy Cards + the journey(s) in one transaction, you pay and the machine will issue a card with the journey(s) loaded on to it.
Though some machines only allow pre-purchased Navgio Cards to be inserted and loaded.
Of if the staffed ticket counter is open (see below), you can obtain the Navigo Easy cards + the journeys in one transaction at the counter.

Once you have the Navigo Easy card keep it in your card holder and you can then use it when you return to Paris.
At some busy stations you may see staff on the concourse holding mini payment machines, and when you already have the Navigo card, you can approach these staff and add tickets / journeys to the card.

Using the apps

The big plus of the apps is that because you can use mobile banking to pay for transactions the ticket machines and counters at the stations can be avoided.
Post-purchase, you can tap your phone on the card readers on the ticket gates to open them.
Though you need an app / mobile device per person, you cannot purchase tickets / journeys for more than one individual within the app.

However, there are two apps:

1: The Bonjour RATP app keeps things simple and it's the best option if all you'll want to do is purchase the ticket for the Metro journey to the Gare Du Nord
Because the advantage of using it is that you don't have to bother with the Navigo Cards / Passes, as you can buy a ticket(s) online and store them within the app.
Plus it's typically possible to make the purchase(s) in the app before you get to Paris.

2: - Using the Île-de-France Mobilités app - aka the Navigo Pass app

If you have a compatible iPhone or Apple Watch this app is built into the Apple Wallet
You can add it by tapping the Add (+) button in the Wallet app, selecting "Transit Card," and choosing "Navigo".

Though as explained in depth on this excellent guide, downloading the Île-de-France Mobilités app and setting it up for purchase is a multi-step process if you have an Android device - and the phone needs to be NFC compatible.

However, it seems as though you need to be in Paris before you can make any transactions / payments on the app.
So the best methodology seems to be (1) download/ open the app before you set off, then (2) buy the ticket(s) on the app when you get to Paris.

The key difference to the Bonjour Ratp is that you can use an Île-de-France Mobilités app to purchase a range of passes aka 'titles' in addition to the Zone 1-5 tickets.
So it can be particularly worth the bother if you know that you'll be likely to return to Paris for an extended stay.

The excellent parisbytrain website has all the info you will need about metro tickets.

The Line 4 trains depart from the part of Montparnasse – Bienvenüe metro station that is located furthest away from the entrance to the Metro.
The longest passage way between the entrance to the Metro and Line 4 houses a long travellator.

At the end of the travellator look out for the Line 4 signs to trains heading to Porte de Clignancort, don’t follow the signs for trains heading to Marie de Montrouge.

You will need to go down a set of stairs to access the platform/voie which the trains heading to Porte de Clignancort will depart from; there is no escalator or lift access.

Having exited the Metro station there are two routes available into the station, heading to the left or going straight ahead.
If you head left down the passage way, after a minute on the right you will see stairs and an escalator, which can travelling up or down, this exit from the passage way leads up to the concourse which most of the long-distance trains use.

From this exit, over to the right will be the voies (platforms/tracks) which the Thalys trains depart from, so if you follow the signs from the metro exit, which point to the Thalys trains, you'll be using this access to the concourse.

Most of the TGV trains to other destinations in France also leave from this part of Gare Du Nord, as do the Eurostar trains, but the access to Eurostar departures is in a different part of the station, which is above this link between the Metro and the concourse, but there is alternative, more obvious route available.

In this view taken from above, the exit from the Metro station is in the far left corner, to the right of the blue poster - the escalator in the middle of the picture is the main link up to the main departure concourse.
You will exit into an atrium, pictured below, from where you will see the main concourse at the Gare Du Nord above you.

Take the escalators or lifts up to the main concourse; so if you follow this route and everything is in working order, you will have step-free access from the metro exit to the station concourse which is where the mainline trains depart from.

Walk ahead of you to access the part of the concourse adjacent to where the TGVs and Thalys trains depart from, you'll see them ahead over to the right.
Because the trains are over to the right, it's easy to assume that you'll need to head towards them, but you will have to go up another level to access Eurostar departures; the ‘Hall du Londres’ and the escalators and lifts (ascenseur) leading up to it are over to the left as you walk towards the trains.
They are located on the opposite side of the concourse to voies/platforms/tracks 14 - 15; look out for the Union Jack on the rear wall of the station concourse.

However, if you are connecting to the Eurostar and have Standard or Standard Premiere tickets, Eurostar is currently recommending tghat you should be at the entrance to the Hall Du Londres 90-120 mins before departure; and that Business Premier ticket holders should be there 45-60mins prior to departure.
The departure gates will close 45 mins prior to departure for Standard Class And Standard Premier ticket holders; and 15 mins prior to departure for Business Premier ticket holders.