Played with STB sharpness from 1 to 5. No significant changes.
We used to have just one TV, but all this upgrade-to-Quantum story started with adding another TV. Verizon technician also suggested to try this second one (which is a brand new). He admitted that on the newer TV the picture looked even worse...
Will continue troubleshooting process (without some serious hopes, though...). Will try to remove a coax coupler (from the wall outlet) and connect the cable directly to the STB (without the coupler and another short piece of cable that is currently goes out of the outlet), etc. Probably at some point will find a weak link... The final test would be direct ONT - STB connection (no splitters, no couplers, etc.). If that does not work will have to do something crazy, such as leave Verizon Internet (which is perfect) and try some Direct TV...