No, they don't track that information. The chips in the ink carts have a date, and I think a fuse or set bit for low ink.
The older inkjets, 800 and 900 series I think, had a 4-bit number that was read from 4 pads on the cart.
The printer would “remember” the last 4 carts it had seen, and give an error if that cart was marked empty in printer memory.
We used to put bits of tape over the pads to change their digital readout, and reinsert them 4 times to clear the memory, then the printer would see them as new.
You can get replacement chips for newer carts, or get refillable cartridges. HP really doesn't like aftermarket ink and does what they can to discourage you from using them.
If you send the printer in with aftermarket non-HP carts, I think they can void your warranty. Some HP inkjets have a separate replaceable printhead that can get clogged.
There are methods to clean, and unclog them that you can try. The printheads are not super expensive but do often clog if seldom used, in hot weather, or cheap aftermarket inks are used.
I suppose they could theoretically test the residual ink for the aftermarket, but I doubt they will do so. Have seen an error message about counterfeit carts used. 
A factory reset should clear that if you can do it.
Obviously, don't take it in with non-HP carts in it. They will use any excuse to invalidate your warranty if they can.
Often the cost of repair is as much as buying a new printer, as they sell their consumer grade printers as loss leaders to keep you buying their horribly overpriced ink refills.
You need of any our Technical Support agent. Please call Technical Support team at +1 888 597 3962 our experts are available 24/7 by phone.