Sounds like something I've experienced before but sometimes symptoms can be due to a few reasons and not just one so I'll give you some of my feedback on what it could possibly be as I own a 2012 FX35 AWD and have done quite some maintenance on it. You haven't listed the mileage but my basis is going to be regardless of mileage (but please do tell when you get a chance as it helps). Through my experiences with the FX35/QX70, they are amazing vehicles but they also like to have their maintenance a specific way but nothing too crazy.
So here's what I recommend.
1. Depending on the mileage, if its nearing 100K miles or over, change the spark plugs with the spark plugs they sell at the Infiniti dealership. They're usually NGK's. Don't get too fancy with Bosch or any other brands elsewhere. There's a lot of fake knockoffs floating around the internet, amazon, etc, even at stores so a few more bucks will save you from a destroyed motor over a simple spark plug. You don't have to get them replaced by the dealership. These high performance engines chew up spark plugs and they should be replaced at regular intervals of 100K miles. If they are prolonged past 100K miles you'll end up with misfires and low power.
2. Coil packs usually don't fail but sometimes they do. I've never had one fail on an Infiniti so I'd leave this alone for now. Change the spark plugs first and see what happens. Coil packs aren't cheap but if you replace them, its recommended they are replaced in sets. So 6 coils packs and OEM, no aftermarket or OE grade crap.
3. Change the transmission oil. With what you are describing with the gear kick, it sounds like the transmission oil needs a change. This should be changed every 55K miles as per the transmission manufacturer. Infiniti states not to change it ever which is a bunch of BS. That's just politics where Infiniti wants to prove to the government that their vehicles are "greener" and don't require much fluid changes especially hydraulic oils like ATF. Many automakers are in the same boat, ignore this rule, there's no such thing as forever oil and get it changed at regular intervals. I use Motul ATF VI which is recommend highly in the forums and works well with Japanese transmissions. One thing I also recommend is when replacing the transmission oil, before the old oil is drained, have them pour in a can of Motul transmission cleaner fluid in the transmission first and with the vehicle stationary, shift through all the gears (P -> R -> D -> R -> P) a few seconds in each gear to clean all the passages for each gear. Have them do this a few times and then replace the transmission oil. Under no circumstances have the vehicle be driven with the transmission cleaner in there!!! The cleaner basically removes any varnish that has built up in the passages inside the transmission especially the solenoids. Varnish tends to cause the valves in the transmission to stick. One thing I did was before I replaced the transmission oil, after I did the motul cleaner, I drained the oil from the transmission oil pan, dropped the transmission oil pan, cleaned the oil pan and magnets, then installed the oil pan with a new gasket (make sure its new or it will leak and only OEM), then I poured in new transmission oil in equivalent what I drained. Now the way I did it was different than what a shop will do. A shop will hook it up to a special machine that will put in new oil while pumping out the old. If they have such a machine, they can add the transmission cleaner to the machine and then shift through all the gears a few times before they replace all the oil.
4. Check the engine air filters. See if they are clean or relatively clean. If you have K&N filters, see if they are relatively clean and not over oiled. If they are over oiled then there's a chance that the MAF sensors (just upsteam from the air filters) are coated with oil and are measuring the air incorrectly. This can be easily fixed by removing them from the air intake tube (2 screws), disconnect the harness, and spray them with some MAF sensor cleaner spray. I recommend CRC but there's other brands that are good as well. I spray the inside a few times and let it drain out. The MAF cleaner is quick dry so it should be dry in 5 - 10 minutes. You can also used compressed air to dry the MAF sensor faster. Just hold the MAF sensor well so it doesn't fly out of your hand and hit the ground. They are sensitive fragile sensors so if it falls it might be toast. If the air filter is a K&N and is over oiled, have them washed and re-oiled after air dried (naturally). I buy the K&N kit with the spray oil and spray cleaner. I spray the oil on the side where the air comes into the airbox, never the side that faces the MAF sensor/engine. I spray the filter twice. Once vertically and once horizontally. That way there's an overlap. Then let it soak for 30 minutes then lay the filter down on a piece of paper towel so any excess oil is wicked out. That way you won't have that oil coating the MAF sensor.
5. Clean the air intake throttle bodies. This one should be done I would say along with the transmission oil change and MAF sensor cleaning. This does help and usually resolves a few issues that surround how the engine responds with the transmission. After a cleaning you'll find that the car drives much nicer. Why this is good to do is because the crankcase oil vapors are sent into the intake where it passes through the intake throttle bodies and over time they get coated with oil. This oil eventually causes the throttle body "butterfly valve" to gum up so they don't quite respond as quickly as they should during shifts and what not. CRC makes a spray for this specifically as well (there's other brands too). What I do is take off the intake tubes, then the intake throttle body. I don't disconnect the harness off the throttle body simply because the computer might trigger a code/fault that will require a relearn. Plus there's coolant lines that go through the intake throttle body and its not necessary to go through all that trouble for a simple cleaning. I take a shop towel, preferably a few or a thick one. Lay it in your hand and place the throttle body on top of the shop towel in your hand. Then spray the throttle body cleaning on the butterfly valve just enough to wet it. Then if you have a brush, like a round painter brush, something as thick as a finger for size, and brush around the butterfly valve agitating the dirt/oil. You can spray more of the throttle body cleaner to wash off the dirt. Repeat this until its all gone on one side of the throttle body. Make sure to wipe off the throttle body cleaner on the clean side and then gently turn the throttle body over so you can see the other side of the throttle body and do the same procedure. Spray, brush with the brush until clean. This will likely push dirt back to the other side. That's okay because at this point you'll spray more of the throttle body cleaner on this side until all the dirt goes away. Might be good to have some extra shop towels handy in case the ones you have in your hand soak up and get too wet. Once that's clean, just re-install the throttle body and do the other side. Just a note, I've seen people open the butterfly valve with their fingers and clean around the edges. I would avoid doing that simply because I've seen people break their throttle body and they are expensive. Some people have somehow gotten away with it but like I always say, I rather be 95% clean than 100% poorer in that sense. The brush will get into where it needs to and the rest will dissolve with the throttle body cleaner. The throttle body cleaner also quickly evaporates.
6. Reset the ECU. There's numerous ways to do this. I do it with a bluetooth OBD2 dongle I connect to the OBD2 port but there's ways to do it by disconnecting the battery. Or you mechanic can do it as well. This is always good to do so the car relearns everything especially now that things have been changed/cleaned.
Once you go through the above steps your car should be much better than before if not like new. It's usually not easy to diagnose a problem without being able to see if for yourself but overall, anytime I buy a car used, I replace all the fluids on it so I know it has the right spec quality fluids. Spark plugs, battery, air filter, oil filter and oil. You'd be surprised how many people cheap out or put in the wrong stuff just because they think it doesn't matter, but it does. Everything comes down to a science and these cars in order to perform correctly must have what they need.
I hope this helps you out, but if it doesn't, the journey will continue so message back and we'll take it to the next stage after this.
Happy New Year!!!