Back in the days of sailing ships, swords and muskets, victors in
combat would generally provide prisoners with shelter. To grant
quarter, as it was called, was an act of mercy.
Pirates who roamed the seas flew a flag bearing the skull and
crossbones to announce they would take no prisoners - they would grant
no quarter. The defiant defenders of The Alamo made it clear they would
fight to the death - they would ask no quarter.
In modern lacrosse, "no quarter" means sweating blood all winter so
you can dominate in the spring.
It means sprinting to the cage when the long stick is gasping.
It means playing defense so tight the attackman can't spit, let alone
get off a shot.
It means being first to every ground ball and putting your face near
the dirt when you get there.