I would like to counter the naysayers. As long as we're dreaming of terraforming, this seems as good an idea as any.
First, I think there's a bit too much hand-waving along the lines of, "Well, once we could do that, we might as well... do... something... else??"
That begs the question, "What else?"
Two counter-suggestions, namely, using the elements in the Martian crust and using comets and asteroids each bear their own consideration.
Second, I think that there's a lot of heft to the idea of terraforming our solar system as a tandem problem, wherein any solution to the problems of one world may best be addressed by solving an inverse problem on another world. Mars is too cold and its atmosphere too thin. Venus is too warm and its atmosphere too thick.
Still, let's consider the alternatives in depth.
Using the Martian Crust
Considering the elemental composition of Mars's crust is important, but NASA-sponsored research has suggested that there likely is not enough CO2 in Mars's ice caps and crust combined to pressurize the atmosphere for a greenhouse effect (source).
Likewise, the same paper cited in the comments doesn't indicate an abundance of nitrogen in the Martian crust that we could use for terraforming. Certainly any serious terraforming effort would require us to maximize the use of resources "on the ground", but if those resources aren't in the ground we have only one alternative: we will have to go somewhere else for some of our elemental needs to make Mars livable.
Comets and Asteroids
This is likely to be an important piece of the puzzle, in no small part because it's feasible with today's technology. We know how to land spacecraft on comets and asteroids. Likewise, we could conceivably produce rockets with enough thrust to push comets and asteroids into an impact orbit with Mars.
Further, comets and asteroids have lots of desirable materials relevant to terraforming. Comets in particular are rich in water, CO2, methane, and ammonia, which finally gets us the nitrogen from the original question.
My chief question is what would be the actual effort required to wrangle enough comets and asteroids to get Mars's atmosphere up to an Earth-like atmospheric pressure?
There probably is enough raw material in the various asteroids and comets around the solar system to bring the Martian atmosphere up to one atmosphere. Indeed, the NASA paper cited above notes this.
Another idea is to import volatiles by redirecting comets and asteroids to hit Mars. However, the team’s calculations reveal that many thousands would be required; again, not very practical.
So, the question is:
Is it logistically more prohibitive to put enough comet-chasing rockets into space and then turn them all to the purpose of redirecting the orbits of various small bodies than it would be to cart Venusian atmosphere to Mars?
I'm not sure, but let us at least consider the alternative.
Back to Venus
One of the chief objections here is the cost of moving the atmosphere out of Venus's gravity well. Yet this is a central problem to most multi-planetary-scale engineering problems with lots of different proposed solutions. I would like to draw attention to the humble skyhook. Sure, it's not as flashy as a space fountain or space elevator, but it could be perfect for mining the Venusian atmosphere at scale.
On its descent into the atmosphere, one end of the skyhook could begin pumping material out.
On ascent, it could release it into a higher orbit where it could be gathered up and pressurized by an orbital tanker.
From there, it could be shipped off to Mars.
Granted, pressurizing a quadrillion tons of atmosphere and shipping it across the solar system is no mean feat.
And yes, a skyhook for Venus would be the single greatest engineering feat our species has yet accomplished.
Of course, so would putting thousands of comet-moving rockets into deep space.
The question is, what are the relative pros and cons of each approach?
The big advantage to Venus is that we begin the process of thinning Venus's atmosphere, which is an important terraforming project in its own right.
Another advantage I see over comet mining is that it's just one big problem to worry about, rather than thousands of small ones.