Marvel has not given a canon explanation why Quicksilver does not create sonic booms when he moves faster than sound.
SPECULATION BASED ON OBSERVATIONS
Canon Continuity Earth-616
Quicksilver, Pietro Maximoff was a mutant and the son of the near-Omega class mutant, Magneto. He is also the brother of the mutant sorceress, the Scarlet Witch. His x-gene adaptations gave him the ability to run at incredible rates of speed.
They adaptations included:
A degree of superhuman strength (mostly in his lower body) and incredible stamina, a physiology capable of sustaining his hyper-fast movement due to nearly frictionless joints and a resilient skeletal structure. (In all the years, we have see Quicksilver fight, he rarely breaks a bone even when he is moving as super-speed.)
He was also able to heal faster due to his accelerated metabolism.
His brain was also accelerated, increasing his thinking processes and giving him the perceptual abilities to think in conjunction with using his power.
His power was written (as much as could be expected) to make sense as a adaptation or evolution with a plausibly-deniable sense of physics and science.
This meant back in the day, his top speed was the speed of sound. He didn't make sonic booms because he needed quite a bit of room to run and he could barely sustain that speed for long.
And yes, he would make a sonic event, but it wouldn't be called a boom, because he was not generating very much sound. Part of the sound of sonic booms comes from the engine sounds being compressed along the wing and nose of an aircraft.
He was also quite material when he ran, he didn't have the power to phase through objects back then. Note his fight with Spider-Man and how Spidey ends the fight.
Pietro has experienced significant power increases in recent years, with a corresponding lack of physics integrity.
These days, Quicksilver's power has graduated quite a bit. He is taking pages from DC's Flash, including running much faster than before (his top speed is now Mach 5).
He is immune to the friction and any impacts taken while he utilizes his speed. He is able to run across the surface of the ocean or up the sides of buildings with ease.
He is capable of molecular phasing: The power to control his molecular movement and pass through solid objects is the same power that makes him immune to friction and impact while running up to speeds of Mach 5.
Molecular phasing would also be responsible for reducing his creation of sonic boom when moving faster than sound and avoiding damage, since he can be at least a bit out-of-phase while running, there is no pressure wave building up in front of him, compressing until he passes the speed of sound.
Thus, no sonic boom.
Here is another canon appearance where he is moving at speeds far beyond the speed of sound without a corresponding boom.
