This is one of those "it depends" kind of questions. But it's really a pretty good one.
"Mars has a valuable resource that Earth wants and therefor it funds the colonization effort. Mars wants trade as equals rather than being a colony that must pay tribute to the motherland."
This puts us squarely into Imperial territory: the concept of the "wealth pump". An empire exerts control over a foreign territory in order to extract wealth in some form. In this instance, it's defined as a single resource.
It's kind of simplistic, but probably useful for present purposes, to say that there are are four main methods of imperial control:
Force: direct military action. This is pretty obvious. Historical examples abound. Given the difficulties of interplanetary combat, it might not enter into the picture here.
Force: withholding necessities. This one is much less common in Earth's history, but clearly a factor here, when Earth can simply stop sending basic supplies that are necessary for the Mars Colony's wellbeing.
Financial compulsion. This is the primary technique that the present-day American empire uses: legal and monetary systems of unequal exchange, enforced via such instruments as transnational banking systems and the International Monetary Fund. I mention it because I'm sure that the Earth - Mars conflict you are seeing would involve legal and financial arrangements of this sort.
Religious or ideological control. You don't mention this one, and it probably wouldn't make too much sense in the context. However, the longer the timespan we're looking at, the more likely systems of ideology are to grow up around the situation.
To answer your question: How long would it take for the Mars Colony to become able to defy the authority of Imperial Earth?, you need to first answer the corollary: How do the Martians get it done? This is in turn worked out by analyzing Earth's control over Mars.
The Martian goal is "trade as equals, rather than being a colony that must pay tribute". This means that Mars has the initiative: until the Martians do something, Earth will be likely to remain satisfied with the situation it has already set up.
To dig a little more into the Martian goal, here are some characteristics that would be necessary:
The four means of imperial governance noted above must be collectively more expensive than they are worth. To take the military example: the cost of interplanetary military expeditions would be very high, even with the reactionless space drive you posit. When does someone in charge on Earth stand up and say, "Why do we keep blowing money out of the airlocks? Can we get the stuff some other way?"
None of the four means of imperial governance can be overwhelmingly decisive. For example, if Mars depends on shipments of water all the way from Earth, Mars isn't going to be in much of a position to do anything about the situation. So you need to work out how the Martians will be able to sustain an independent existence.
Earth's desire for the Martian resource must be high enough to provide a strong incentive to come to terms with Mars; but not so high that Earth is willing to pay enormous prices in order to get as much as desired.
So, there is the Martians' desired strategic end state.
How long will it take the Martians to get it done?
In my opinion, you have enough flexibility built into your world here that you can pretty much make it take as long, or as short, a time as you please. Between your ability to modify social, economic, and political conditions on Earth; your ability to tweak the characteristics of your space drive and your precious Martian resource; and your array of Mars Colony scenarios of population and industrial capacity; you could easily set this up to run long or short.
Up to you. Your world will support these outcomes.
Honestly, now you get the fun part: playing with different parameters to set up a piece of invented future history that is intellectually absorbing, and satisfyingly dramatic. :-)