Legendary Pokemon have low catch rates, and require you to get them
- to a low amount of HP (Preferably in the red)
- have a non-volatile status (Poison/Sleep/Paralysis/Burn, of which Sleep is most effective)
Thus, these require multiple turns.
The thing about Quick Balls is that they are very useful the very first turn, and keep getting progressively worse, until they become as effective as normal Pokeballs by around turn 5. They are thus recommended for low level, easy to catch (high catch rate) Pokemon more than Legendary ones like Regice.
Thus, your best bet is to lower its HP, put it to sleep, and spam Ultraballs.
If you're lucky, though, using a Quick Ball the first turn might let you catch it. If not, try the other steps from turn two onwards and spam Ultraballs, which remain equally effective no matter what turn it is.