No a trespassing warning will not show up on your record.
When a person is trespassing and someone calls the cops or the cops see you trespassing on the property they will first warn you of the trespassing.
Then if you return to the property when you've been warned of trespassing then the second time you're caught you'll be arrested.
After the second time you're trespassing and you're arrested then the trespassing arrest will show up on your record.
But just a trespassing warning will not show up on your record even though the police will take your ID and fill out a form about you being warned of trespassing.