Can Improvise pay for "twobrid" costs?
As part of casting a spell, you must announce the nonhybrid equivalent cost you intend to pay.[601.2b] For Beseech the Queen, this is any of the following:

{6}
{4}{B}
{2}{B}{B}
{B}{B}{B}

Each artifact you tap covers {1} of the cost, so yes, you can pay for Beseech the Queen by paying {1}{B}{B} and tapping an artifact if you first pick {2}{B}{B}.


601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. Casting a spell includes proposal of the spell (rules 601.2a–d) and determination and payment of costs (rules 601.2f–h). To cast a spell, a player follows the steps listed below, in order. [...]
601.2b [...] If a cost that will be paid as the spell is being cast includes hybrid mana symbols, the player announces the nonhybrid equivalent cost they intend to pay. [...]
702.125a Improvise is a static ability that functions while the spell with improvise is on the stack. “Improvise” means “For each generic mana in this spell’s total cost, you may tap an untapped artifact you control rather than pay that mana.”
702.125b The improvise ability isn’t an additional or alternative cost and applies only after the total cost of the spell with improvise is determined.