Your question may be divided into three parts.
The Plate tectonics but no earthquakes part has already some good answers, I will focus on the other two parts.
Experiencing zero or weak / little earthquakes
I experience weak earthquakes every day, when the next freight train passes within 50m of my flat. The several natural earthquakes a month in this region are to week to distinguish them from the footsteps of my neighbor. If there were no stories about it I would not know that natural earthquakes exist. This is not true for every place on this planet, but it should be for some of them.
Science can only provide probabilities for the next earthquake. Theoretically it is possible that whole generations over the whole planet do not experience any not-weak earthquake in their lifetime.
So in a given time in a given place there are no stronger than weak earthquakes even on earth, so it is possible.
Maintaining an environment habitable for life without earthquakes
As I understand the question, you try to maintain an ecosphere without regeneration of critical elements through subduction and volcanoes.
The answer is easy, it depends on your planets design. Elements like e.g. iron, hydrogen, oxygen and nitrogen have to be somehow introduced to the ecosphere and/or regenerated from deposits like sediments. Introduction might be accomplished through meteorite impacts or better some space dust (we don't want more earthquakes). If you need to regenerate more than through the normal biogeochemical cycles you might need some natural or artificial method to get sediments from the ocean floor and put them on higher ground like the bear-salmon-phosphor-pump, a combination of shallow water and strong seasonal winds (sadly no tides because of the earthquakes) or mangroves everywhere. I expect biology to adapt to smaller amounts of problematic elements, at least for some time, too, like the amazonas rainforest.
I could try to calculate how many years this ecosystem would be in a more or less stable flow equilibrium (meaning: individual inhabitants don't experience a change), but there are too many unknown variables here and even the research on this problem looks more like our nitrogen comes for several billion years out of plate tectonics and volcanoes so mars and venus are loosers than how long it would last without or how much it changed in the last million years. Just enter whatever time you need between some hundred and some billion years or put down some more of these meteorites before the time anyone can remember (because of the earthquakes of course).
I hope this helps.