When shifting gears car vibrates, more so when accelerating

I'm sorry to hear that. The Note uses a CVT transmission as shown below. They are very problematic and seem to fail far more often than they should. It is one of the reasons why other companies tried them and then stopped using them shortly after. Vibration and roughness during shifting can be a sign of the drive belt inside the CVT starting to fail.

However another item that could cause a shake when you increase the load on the engine would be an intermittent misfire, with the 3 cylinder engine there isn't any extra power pulses to help hide them. Is the check engine light on or has it ever flashed while the shaking was happening? If not then it is likely the CVT starting to fail.
There have been a few updates to the software that are supposed to help improve the transmission, you might want to see if those have been applied, a dealer would have done them and should be able to look up the VIN number to see if they are there.

It would depend on what the code was, however if it hasn't come back in that time it may have been a fluke. The light will come on if there is something that calls for it. Unfortunately there are many codes that don't turn the light on, but are still there and a dealer level scan tool can read them. Then there are systems that can have a fault and never set a code because they are not monitored by the computer. For instance brake wear, transmission fluid level or quality, or if the item does set a code it may take more than one failure to turn on the light. Those can be hard to track down.

I would probably start by using a scan tool to see if there are any hidden power-train codes. Then go from there. In my area the process is usually to drain and examine the fluid, looking for anything indicating internal damage like metallic bits or flakes of belt or sheave material. Then it becomes a choice of repair or replace. The drive belt in that transmission isn't an easily replaceable part, most of the shops and dealers around here don't even open them up, they remove the bad one, put in a new or rebuilt unit and send the old one back to the factory to be rebuilt. There are a few transmission shops that will rebuild them but it depends on the parts availability and what the shops have tooling to do. You mentioned that it started doing it a couple months after you bought it, I wonder if the seller had the transmission serviced and they used the wrong fluid in it? Either way a fluid change shouldn't hurt it and you would need to drain some out just to see if there were debris in the current fluid.

I mention the servicing because there have been issues with some places that do fast oil and transmission fluid changes who use a universal base fluid that they add a modifier into that magically turns it into the "correct" fluid for whatever transmission they used it in. In many old style automatics this isn't a huge thing because they are really all very similar in the internal parts so as long as the fluid was close it would work alright. These days however close doesn't count, especially with a CVT, the belt inside them generates a lot of friction and heat and the fluid used pulls the heat away and lubricates the surfaces of the sheaves as the belt moves up and down on them.
Keep us in the loop as to what happens please.

Thank you for the update and thank you for using 2CarPros.