As written, booming blade doesn’t make any more noise than a normal attack.
Notably, booming blade does not have a verbal component, and the spell description says nothing about making noise. Compare it to the spell thunder step:
The thunder can be heard from up to 300 feet away.
Thunder step makes extra noise, because it says it makes extra noise.
It is worth mentioning that the description of thunder damage does involve sound:
Thunder. A concussive burst of sound, such as the effect of the thunderwave spell, deals thunder damage.
But this does not tell us the magnitude of this burst of sound. Thunder step tells us it is much louder than a normal attack, so it is, but we have no reason to believe that the sound made by booming blade is a particularly loud sound, not any louder than we would expect a normal attack to be.
If your DM is ruling that booming blade makes noise because Thunder damage, use green-flame blade instead.
Green-flame blade is booming blade’s twin brother, and has almost the same effect (secondary damage is to a second target, rather than the original target), and deals fire damage instead of Thunder damage.
Unfortunately, there really isn’t much else to do here. No feats give access to the silence spell.
Given its name, you would think Subtle Spell would work, bit it won't.
Unfortunately, Subtle Spell sounds like it might be helpful here, but it's not. Subtle Spell, the metamagic from the Sorcerer class, only removes components required for casting a spell:
When you cast a spell, you can spend 1 sorcery point to cast it without any somatic or verbal components.
Don't mind me while I subtly cast thunder step.
Guy 300 feet away: Wow, that was not subtle.
Take the Metamagic Adept feat and select Transmuted Spell
Tasha's Cauldron of Everything introduces a new feat and a new metamagic option, and as user findusl pointed out in a comment, they may help here. Metamagic Adept states:
You learn two Metamagic options of your choice from the sorcerer class. You can use only one Metamagic option on a spell when you cast it, unless the option says otherwise. Whenever you reach a level that grants the Ability Score Improvement feature, you can replace one of these Metamagic options with another one from the sorcerer class.
With this, you can select the Transmuted Spell metamagic:
When you cast a spell that deals a type of damage from the following list, you can spend 1 sorcery point to change that damage type to one of the other listed types: acid, cold, fire, lightning, poison, thunder.
If your DM is ruling that the thunder damage of booming blade makes it exceptionally loud, this would allow you to change the damage type to another, less audible, damage type.
But I must emphasize, this is a really high price to pay to make your booming blade quieter a couple times a day. Forsaking an ASI or a different feat in order to gain this feature is probably far too high of a price. But it's there if you want to pay it.
