I’m going to assume you meant to say “equilibrium quantity” because supply is a curve that dictates equilibrium. The changes in supply and demand will disrupt and shift equilibrium price and quantity, but there is no “equilibrium supply” to increase.
When supply goes down, the equilibrium will change. Price will go up because of scarcity of a good. However, quantity will go down, because with less available, there is a lower quantity purchased. Conversely, if supply increases, price will decrease, but supply will increase.
If demand decreases, the price and quantity will both decrease. If both price and quantity are to increase, there would need to be an increase in demand. This can be driven by better marketing, a change in the market itself, a change in the technology used, or something else—it is a very dynamic construct. However, the underlying economic notion is simple: if you want quantity and price to increase, demand must increase.